588 


Very 

and  Terrestrial 

Albedoes 


I 


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SCIENTIFIC  PAPERS 

THE 

WESTWOOD  ASTSOPHYSICAL  OBSERVATORY 

UMKR    I. 


LUNAR  AND  TERRESTRIAL 
ALBEDOES 


By 
FRANK  W.  VERY 


-  THE  FOUR         MPANY 


OCCASIONAL  SCIENTIFIC  PAPERS 

OF  THE 
WESTWOOD  ASTROPHYSICAL  OBSERVATORY 

NUMBER    I. 


LUNAR  AND  TERRESTRIAL  ALBEDOES 


By 
FRANK  W.  VERY 


BOSTON 

THE  FOUR  SEAS  COMPANY 
1917 


HISTORICAL  NOTE 

FOUNDED  in  1906,  the  Westw'ood  Astrophysical  Observatory  owes 
its  inception  to  aid  from  Percival  Lowell.  In  beginning  a  special 
series  of  its  publications,  the  writer  wishes  to  place  on  record  his 
indebtedness  to  the  warm  sympathy  and  encouragement  of  a  faith- 
ful friend.  Himself  an  ardent  lover  of  freedom,  Dr.  Lowell  never 
interfered  with  the  writer's  free  and  independent  ordering  of  the 
researches  conducted  at  Westwood,  but  with  a  rare  disinterested- 
ness he  placed  at  his  disposal  numerous  spectrograms  taken  at  the 
Lowell  Observatory  by  the  skilful  hands  of  Dr.  V.  M.  Slipher  for 
measurement  with  apparatus  of  the  writer's. design.  Nevertheless, 
the  gain  was  mutual,  for  the  results  throw  unexpected  light  on 
some  of  Lowell's  own  researches  and  demonstrate  that  complete 
independence  in  respect  to  control  and  motives  of  action  is  not  in- 
compatible with  a  consistent  working  together  for  a  common  end. 

Lowell  had  been  greatly  interested  in  the  research  which 
forms  the  subject  of  the  present  communication,  with  its  obvious 
bearing  on  the  problem  of  planetary  temperature.  In  his  "Tem- 
perature of  Mars,"  he  had  adopted  0.75  for  the  albedo  of  a  half 
clouded  earth,  and  I,  in  my  "Greenhouse  Theory  and  Planetary 
Temperature/'  had  taken  0.70  for  the  same  datum,  differing  but 
little  from  the  value  now  found  for  the  geometrical  albedo  of  the 
earth,  which  is  0.72. 

Let  me  also  place  on  record  as  a  result  of  my  intimate  associa- 
tion with  him,  my  recognition  of  the  fact  that  his  theories  were 
based  on  an  elaborate  accumulation  of  unsurpassed  evidence,  that 
he  was  always  open-minded  to  new  evidence,  and  that,  while  pre- 
senting some  revolutionary  new  conceptions,  he  did  not  hesitate  to 
modify  his  own  ideas  when  convinced  that  they  could  be  im- 
proved. It  is  this  Willingness  to  revise  that  constitutes  the  true 
man  of  science.  That  there  was  very  little  for  him  to  change,  as 
his  researches  progressed,  is  a  testimony  to  Lowell's  thoroughness 
and  to  his  deep  insight  into  nature's  mysteries. 

With  gratitude  to  God  for  the  gift  of  a  friend — generous, 
thoughtful  for  others,  and  noble  in  his  ideals,  keenly  critical,  but 
kindly  appreciative,  learned,  but  modest — 

I  dedicate  these  researches 
an  the  Jflrmnrij  of 

Jlernual 


9176: 


THE  WESTWOOD   ASTROPHYSICAL   OBSERVATORY 


THE  WESTWOOD  ASTROPHYSICAL  OBSERVATORY  is  situated  in 
Westwood,  Massachusetts.  Its  approximate  position  and  altitude 
(derived  from  the  topographical  map  of  the  United  States  Geolog- 
ical Survey)  are 

Latitude  =  42°  12'  58"  North. 
Longitude  =  71°  1 1'  58"  West. 
Altitude  =  190  feet  above  sea  level. 

Its  publications  hitherto  have  been  in  current  scientific  per- 
iodicals, especially,  Lowell  Observatory  Bulletin,  American  Jour- 
nal of  Science,  Astro  physical  Journal,  Science,  Astronomische 
Nachrichten,  and  Bulletin  Astronomique. 

The  Observatory  possesses  special  instruments  for  the  study 
of  solar  radiation  and  atmospheric  transmission,  for  delicate  heat 
measurements,  the  utilization  of  solar  radiation  and  study  of  the 
"greenhouse"  effect,  photometry,  spectral  line  and  band  com- 
parator, etc.  Eor  several  years  it  had  the  use  of  a  fine  silver-on- 
glass  concave  mirror  of  12  inches  aperture  and  10  feet  focal 
length,  which  was  loaned  by  its  maker,  Dr.  J.  A.  Brashear.  The 
mirror  was  used  in  researches  on  the  transmission  of  terrestrial 
radiation  by  the  aqueous  vapor  of  the  atmosphere. 

Special  researches  are  being  actively  prosecuted  on  at- 
mospheric transmission  and  the  solar  constant,  quantitative  mea- 
surements of  the  intensity  of  spectral  lines,  planetary  atmospheres 
and  temperatures,  greenhouse  theory,  contributions  to  the  theory 
of  nebulae  and  novae,  measurements'  of  the  earth's  albedo  and  of 
that  of  the  moon  for  all  parts  of  the  visible  spectrum.  The  latter 
researches  form  the  subject  of  the  present  communication. 


LUNAR  AND  TERRESTRIAL  ALBEDOES 


Introduction. 

THE  word  albedo  (derived  from  the  Latin  albus,  white)  has 
been  used  by  astronomers  to  designate  the  fraction  of  the  sun's 
luminous  rays  reflected  by  a  planet  at  full  phase,  allowance  being 
made  for  the  distances  of  the  planet  from  sun  and  earth  and  for 
the  dimensions  of  the  reflecting  body.  If  the  planet  were  a  smooth 
sphere  with  perfect  specular  reflection,  it  would  be  itself  invis- 
ible, but  would  present  writhin  the  diminutive  limits  of  its  disk  a 
complete  picture  of  the  surrounding  heavens,  distorted  by  spher- 
ical aberration,  but  otherwise  exact ;  and  within  this  image  the 
reflection  of  the  sun  would  surpass  in  brilliancy  all  other  objects, 
shining  like  a  star  at  a  point  on  the  planet's  disk  distant  from  the 
center  by  the  radius  of  the  disk  multiplied  by  the  cosine  of  half 
the  elongation  of  the  planet  from  the  sun.  But  whatever  specular 
surfaces  there  may  be  on  the  planets  of  our  solar  system,  they 
are  of  too  limited  extent  to  be  recognized  as  such ;  and  the  plane- 
tary reflection  of  light  is  to  be  classed  under  the  head  of  a  gen- 
erally diffusive  one,  though  not  necessarily  an  equable  one  in  all 
directions;  and  in  fact  there  are  diversities  in  the  distribution  of 
the  reflected  light  to  different  parts  of  the  sphere  which  must  be 
considered  in  getting  the  phase-curve  of  the  illumination,  and 
which  are  not  entirely  without  influence  even  if  we  confine  our 
attention  to  the  reflection  sent  earthward  at  full  phase,  while  they 
are  vital  to  the  determination  of  the  complete  reflection  to  the 
sphere. 

Since  all  of  the  planets,  except  possibly  some  of  the  smaller 
asteroids,  are  spheroidal  bodies,  it  is  not  necessary  for  purposes 
of  intercomparison  to  refer  their  albedoes  to  the  standard  specific 
reflectivity  of  a  flat  surface;  but  it  is  desirable  to  distinguish 
clearly  between  the  only  thing  which  is  certainly  measurable  in 
most  cases, —  which  is  (i)  the  geometrical  albedo  at  full  phase, 
or  the  amount  of  light  sent  earthwards  at  the  planet's  full  phase, 

5 


6     LUNAR  AND  TERRESTRIAL  ALBEDOES 

compared  with  that  which  would  be  sent  by  a  sphere  of  the  same 
size  and  at  the  same  distance,  which  possesses  "perfect  diffusive 
reflectivity; — and  (2)  that  integration  of  the  reflection  to  the 
entire  sphere,  or  the  spherical  albedo,  whose  determination 
requires  a  knowledge  of  the  phase-law.  This  law  is  very  imper- 
fectly known,  except  in  the  case  of  the  ipoon,  and  hence  there  are 
rival  hypotheses  which  give  more  than  one  kind  of  "spherical" 
albedo.  There  is  even  a  diversity  of  usage  in  regard  to  what  shall 
be  called  the  "geometrical"  albedo,  although  there  need  be  no 
discrepancies  in  the  facts  of  observation  on  which  it  is  based. 
A  very  few  words  will  suffice  to  make  the  fundamental  distinctions 
plain  as  to  their  general  principles ;  but  the  remoter  consequences 
of  the  acceptance  of  the  diverse  points  of  view  lead  to  discussions 
of  some  complexity  whose  complete  unfolding  can  not  be  ex- 
hibited in  the  limits  of  this  paper,  but  enough  will  be  presented  to 
give  an  intelligible  conception  of  the  subject. 

If  we  measure  the  .amount  of  light  received  by  the  eye  from 
the  full  moon,  that  is  to  say,  if  we  find  the  reflection  bf  sunlight 
by  a  spheroidal  surface  to  a  point  (since  the  pupil  of  the  eye  is 
virtually  a  point),  we  shall  get  the  same  value  whether  the  moon 
is  near  the  horizon  or  in  the  zenith  (after  correcting  for  the 
absorption  by  the  earth's  atmosphere)  ;  and  it  seems  natural  to 
take  this  constant  light-quantity  as  the  basis  of  the  geometrical 
albedo  referred  to  a  definite  point  in  space,  comparing  it  with  the 
quantity  of  light  which  would  be  given  if  the  whole  sky  were  filled 
with  moons  of  perfectly  diffusive  reflecting  quality,  and  viewed  by 
turning  the  eye  progressively  to  all  parts  of  the  sky  and  summing 
the  successive  impressions.  This  geometrical  ratio  of  the  reflec- 
tion to  a  point  compared  with  the  perfectly  diffuse  reflection  at 
that  point  from  an  ideal  body  of  the  same  size  and  in  the  same 
situation,  is  the  one  considered  in  this  paper  and  is  what  is  meant 
by  the  geometrical  albedo. 

But  if,  instead  of  this,  we  take  the  illumination  of  an  extended 
surface  by  the  hypothetical  sky  full  of  moons,  it  is  necessary  to 
take  into  account  the  diminution  of  superficial  illumination  from 
those  rays  which  are  at  low  angles  to  the  surface,  and  even  sup- 
posing an  absence  of  atmosphere,  the  surface  illumination  pro- 
duced by  a  sky  full  of  moons  will  only  be  half  as  great  as  the 
sum  of  the  illuminations  supposing  each  moon  to  be  successively 


LUNAR  AND  TERRESTRIAL  ALB  EDO  ES  7 

transported  to  the  zenith.    Thus  the  "surface  illumination"  is  one 
half  of  the  geometrical  albedo. 


FIGURE  i 

Lambert  showed  in  his  "Photometria"  (cap.  II.)  that  if  we 
seek  the  illuminating  power  (L)  of  a  circular  luminary  of  radius 
SR  =  r  (Fig.  i),  whose  center  is  at  any  point  (S)  of  zenith- 
distance  SZ  =  "C,,  upon  a  surface  at  C,  we  may  obtain  L  by  sum- 
ming a  series  of  annuli  concentric  with  S  and  of  radius  SX  =  x, 
where,  if  an  element  (dx,  dq>)  of  the  annulus  is  at  the  angle 
ZSX  =  q>.  from  the  vertical  through  5",  the  area  of  the  element 
is  dx  •  dy  sin  x.  Hence 


dx, 


=  I    I  sin  x  cos 


since  the  illumination  of  the  surface  at  C   varies    in    proportion 
to  cos  £. 

By  spherical  trigonometry,  if  s  is  the  zenith-distance  of  any 
point  X  on  the  annulus, 

cos  z  =  cos  £  cos  x  -f-  sin  £  sin  x.  cos  cp, 
and 


8  LUNAR  AND  TERRESTRIAL  ALB  EDO  ES 

L  =  \    \  dx  •  d<p  sin  x  [cos  £  cos  x  -)-  sin  £  sin  x  cos  qp] 


-'• 


A  first  integration  relatively  to  qp  between  cp  =  o°  and 
tp  =  360°,  or  2ji,  gives 


L  =  I  </.*•  sin  x  •  2n  •  cos  £  cos  #. 

Integrating  this  with  respect  to  .r  from  x  =  o°  to  .r  =  r°, 
L  =  2Jt  cos  t,  I  sin  ;r  cos  .r  t/^r 


I  -  COS  2.Y 
=  2n   COS  ^    (-  -)  =r  Jt  •    COS 

4 


This  gives  for  the  illuminating  power  fZJ  of  the  moon  at 
the  zenith  to  that  of  a  sky  full  of  moons  (L')  upon  an  extended 
surface  at  C,  as  a  first  approximation, 

L  :L'  =  n  cos  o°  (sin2  15'  33")    :  ji  cos  o°  (sin2  90°) 
=  i    :  48,875. 

But  if  we  consider  the  luminous  effect  upon  a  point,  such  as 
the  eye,  or  the  heating  effect  upon  the  bulb  of  a  thermometer 
which  may  'likewise  be  taken  as  a  point,  instead  of  that  commun- 
icated to  an  extended  surface,  then,  neglecting  atmospheric 
absorption,  it  is  necessary  to  find  the  ratio  of  illuminations  by 
taking  the  ratio  of  the  area  of  the  apparent  lunar  disk  to  the 
hemispherical  sky  area,  a  ratio  which  is  half  as  great  as  the 
one  just  given.  For  a  disk  as  small  as  that  of  a  planet,  the  area 
may  be  taken  =  Jt  sin2  Q  •  r2,  where  o  is  the  angular  value  of 
the  radius  of  the  disk  and  r  is  the  distance  of  the  planet.  Com- 
paring this  with  the  area  of  the  hemisphere,  2jir2,  the  latter  ex- 
ceeds the  iormer  in  the  ratio,  2  :  sin2  Q,  which,  for  the  moon's 
semi-diameter,  Q  =  I$'  33",  gives  for  the  ratio  of  the  light  re- 
flected to  a  point  from  the  two  sources, 

97750    :  i 
with  a  similar  degree  of  approximation  to  the  preceding  value. 


LUNAR  AND  TERRESTRIAL  ALB  EDO  ES  g 

The  integration  of  the  total  light  reflected  by  the  illuminated 
hemisphere  of  the  planet  in  all  directions  requires  the  introduction 
of  hypotheses.  The  first  is  Lambert's  hypothesis  of  uniform 
diffuse  reflection,  of  which  the  following  account  is  substantially 
that  of  Mullen 


FIGURE  2 


Consider  a  diffusely  reflecting  surface-element,  ds  (Fig.  2) 
illuminated  under  any  angle  of  incidence,  i.  If  L  is  the  quantity 
of  light  which  falls  normally  on  the  unit  of  surface,  then  ds 
receives  L  ds  cos  i,  of  which  a  certain  fractbn  cL  ds  cos  i  is  re- 
flected normally,  and  in  any  other  direction,  such  as  that  of  the 
emanation  angle  E,  the  light-quantity  dq  =  cL  cos  i  ds  cos  E  is 
reflected,  provided  the  surface  reflects  as  well  at  one  angle  as 
at  another. 

Construct  a  hemisphere  with  radius  i  about  ds  of  which  an 
element  dw  receives  the  fractional  light-quantity 

dQ  =  dq  dw  =  cL  ds  cos  i  cos  E  d(a. 

Then  since  the  element  rfca  has  the  width  dv  sin  E  and  the  height 
</E,  or  the  angular  area  </£  sin  E  dv,  the  total  light-quantity  has  the 
value 


Q  =  cL  ds  cos  i 


Jir/2 


COS  E    sin  E    (/£ 


f 

V     0 


dv, 


io     LUNAR  AND  TERRESTRIAL  ALBEDOES 

JTT  72  /?2JT 

cos  8  sin  E  dt=l/2,  and  I         dv  =  2it, 
0  •       t/     0 

Q  =  L  ds  cos  i  •  A, 

where  the  factor  A  is  a  fraction  which  tells  how  much  of  the 
incoming  light  is  reflected  to  a  hemisphere  of  radius  i.  A,  which 
is  always  smaller  than  i,  is  simply  called  the  albedo  of  the  sub- 
stance by  Lambert. 

From  the  two  equations  for  Q,  it  follows  that  c  =  A/n,  and 
thence  we  have  Lambert's  law  of  illumination  by  diffusely  re- 
flecting substances  in  the  well  known  form  : 

dq=  —  L  ds  cos  i  cos  e, 

Ji 

or 

dq  —  F!  ds  cos  i  cos  e 

if      r  = 


Calculation  of  the  quantity  of  light  sent  to  the  earth  at  dif- 
ferent phases  of  a  reflecting  planet  requires  some  further  slight 
modifications. 

Let  a  plane  be  drawn  through  the  middle  point  of  the  planet 
at  right  angles  to  the  earth-planet  line  ;  and  let  its  intersection 
with  the  planet's  surface  be  represented  by  the  circle  ABCD 
(Fig-  3)-  The  perpendicular  to  this  plane  in  the  direction  of  the 
earth  is  shown  diagrammatically  by  ME.  MS  is  drawn  in  the 
directon  of  the  sun.  The  arc  of  a  great  circle  ES  is  the  phase- 
angle  a,  taken  from  full  phase.  An  element  ds  of  the  visible 
hemisphere  of  the  planet  is  connected  with  E  and  S  by  great 
circles  of  the  sphere.  Arc  S  —  ds  =  i,  arc  E  —  ds=E.  Latitude 
of  <ly  =  \|j  =  F  —  ds.  Longitude  fro'm  E  =  EF  =  (d. 

From  the  right-angled  spherical  triangles  FSds  and  FEds,  we 
have  the  relations  : 

cos  i  =  cos  i|>  cos   (co  —  a), 
cos  E==  cos  ty  cos  (0. 

If  the  semi-diameter  of  the  planet  is  Q,  the  linear  dimensions 
of  ds  are  Q  dty  in  the  direction  of  the  meridian  and  9  rfto  cos  ip 
along  a  parallel.  Hence 

surface  of  ds  =  Q2  cos  ip  <fa>  d\\). 


LUNAR  AND  TERRESTRIAL  ALBEDOES     n 

J 


We  now  introduce  two  rival  hypotheses,  or  laws  of  reflection, 
(i)  that  reflection  is  uniform  in  all  directions,  (2)  that  it  varies 
according  to  a  definite  law,  and  get 


(0 


Lambert's  Law : 

i  a.     dq^  =  r\  o2  cos3  \p  dty  cos  (co —  a)  cos  co  dco, 

Lommel-Seeliger  Law : 

cos  co  cos  (co  —  a) 


ib.     d.  = 


o    cos 


cos  (co— a)  -}-?.  cos  co 


a'co, 


where   P.,  = 


L 


k 


and   X  = 


,  k  being  the  coefficient  of 


absorption  of  the  rays  which  enter  into  the  interior  of  the  sub- 
stance of  the  planet's  surface,  kl  the  coefficient  of  interior  ab- 
sorption of  the  returning  rays  on  the  way  to  emission,  and  p. 
the  diffusive  power  of  the  body.  In  general  £t<£,  because  the 
outgoing  rays  have  lost  their  more  absorbable  ingredients.  If  the 
material  is  strongly  colored,  k  may  be  very  much  larger  than  kr 


12  LUNAR  AND  TERRESTRIAL  ALBEDOES 

The  derivation  of  the  Lommel-Seeliger  equation  which  takes 
account  of  interior  reflection,  and  diffusion  is  very  complicated. 
The  final  equation  is 

cos  i  cos  6 


ds 


cos  i  -(-  A  cos  E 


Confining  attention  here  to  the  Lambert  equation,  the  formula 

must   be   integrated   over   that   part   of   the   illuminated   surface 

visible    from    the    earth.     The    integration    limits     for    ip    are 

-  Jt/2  and  -f-  jt/2,  and  those  for  to  arc  — 11/2  -f  a  and  4-  Jt/2, 

pr/2  pr/2 

whence  <71  =  ri  o2    I  cos3  ip  chp   I  cos  (w  —  a)  cos  co  dto. 

t/       -  IT  /-  t/a-TT/2 


But 


J1T   /  2  A*TT  /  I! 

cos3  \p  (/x))=  I  cos-  ip  fl?(sin 
-7T/2  e/        -7T/2 


=  r 


ip]  c/(sin  \p)  =    -• 
And 


J 


7T/2 

cos  (to  —  a)  cos  to 

a-  TT  /2 


—  a)  afco 


J7T  /  2  S*TT  /  2 

cos  v.  dw  -}-  l/2  I  cos  (20)  —  a 
-7T/2  t/a-TT/2 

r:=  /^2  [  ( Jt  —  a)  cos  a  -f-  sin  a] . 
Therefore 

(2)  q1  =  riQ2  •  2/3  [sin  a  -f-  (n  —  a)  cos  a]. 

For  the  full  phase,  when  sun,  earth  and  planet  stand  in  a  straight 
line,  a— o  and  the  reflected  light  is  g1(o)  =  I^o2  •  2/3:1.  Hence 
we  have  for  the  ratio 

Light  at  phase-angle  a  :   Light  at  full  phase, 

(3)  9iA?i(o)=    [sin  a  -f    (it  — a)    cos  a]. 


LUNAR  AND  TERRESTRIAL  ALB  EDO  ES  13 

A   similar   integration    for   the    Lommel-Seeliger   law   gives 

|~i          o 

(4)       q-=—  —  [i — sin  a/2  tan  a/2  log  cot  a/4]. 

But  for  a=  o,  ^.,(0)  =  (T2o2:fi)/2,  and 

(5)  ^2/</2(0)  — :  T  —  gin  a/2  tan  a/2  l°g  cot  a/4. 

The  distribution  of  light  over  the  apparent  disk  of  the  planet 
varies  according  to  the  adopted  law.  Euler's  law  would  demand 
uniform  light,  except  for  a  narrow  strip  of  sudden  diminution  at 
the  terminator  and  an  excessively  narrow,  but  exceedingly  bright 
rim  at  the  illuminated  limb.  Nothing  of  the  sort  is  observed,  and 
this  law  may  be  dismissed  at  once.  Moreover,  Euler  considered 
nothing  but  superficial  reflection,  just  as  Bouguer  did,  whereas 
the  penetration  of  the  light  into  a  thin  surfa'ce  layer,  even  in  the 
most  opaque  substances,  is  of  great  importance. 

Lambert's  law  appears  to  work  fairly  well  where  the  reflec- 
ting medium  is  of  the  nature  of  cloud  with  internal  diffusion  and 
multiple  reflection  from  innumerable  widely  dispersed  and  finely 
divided  particles,  such  as  ice  crystals,  or  dust,  or  the  liquid  water 
particles  of  ordinary  cloud.  The  Lommel-Seeliger  Law  is  more 
appropriate  for  extended  solid  surfaces  at  various  inclinations  to 
the  incident  light.  A  composite  inter-mingling  of  solid  surface 
and  cloud  requires  a  mixture  of  the  two  laws. 

In  the  following  Table  are  given  the  computed  values  of  the 
functions  of  the  phase-angle,  <p  (a),  for  intervals  of  5°  according 
to  several  theories.  These  quantities  are  then  multiplied  by 
others  proportional  to  the  areas  of  the  corresponding  zones, 
sin  a  •  Aa  =  A(i — cos  a),  to  give  the  values  in  the  last  three 
columns  which,  being  summed,  produce  the  proportional  factors 
for  the  spherical  albedo.  If  the  intervals  had  been  taken  small 
enough, the  sum  of  the  differences  of  versine  a,2  [A  (i — cos  a)], 
would  have  been  exactly  2  which,  multiplied  by  2  JD,  gives  the 
area  of  the  sphere  of  unit  radius.  The  average  reflection  by  a 
planet  to  the  sphere  has  an  intensity  l/±  if  the  reflection  is  perfect 
(normal  specific  reflectivity  =  i),  this  being  the  mean  between 
the  quantities  (l/2  and  o)  sent  in  the  directions  of  source  and 
antipodal  point.  Calling  S  [A  (i  — cos  a)  •  q>  (a)]  the  spherical 
factor,  its  values  given  in  the  last  line  of  the  table,  are 


i4  LUNAR  AND  TERRESTRIAL  ALBEDOES 

0.75  by  Lambert's  Law. 

1.50  by  the  Lommel-Seeliger  Law. 

0.35  by  the  lunar  phase-curve. 

While  the  spherical  albedo  can  not  exceed  unity,  there  may  be 
various  distributions  of  light  to  the  sphere.  Thus,  for  perfect 
reflection,  the  diverse  spherical  factors  obtained  from  the  sum- 
mations in  the  table  are  consistent  with  geometrical  albedoes  of 
0.50,  1.33,  0.67,  and  2.86,  the  last  being  for  the  phase-law  of  the 
moon  where  the  reflection  at  full  phase  is  extraordinarily  large. 


Limits  of 

4 

(a) 

<P  (a) 

en  (a) 

(p(a)  X  A(i  —  cos  a) 

phase-angle 

«!  —  <X2 

Ad-cosa) 

Lambert 

Lommel- 
Seeliger 

i  \  / 

Moon 

Lambert 

Lommel- 
Seeliger 

Moon 

0°—  5° 

.0038 

.996 

.999 

.957 

.0038 

.0038 

.0036 

5  —  10 

.0114 

.986 

.994 

.869 

.0112 

.0113 

.0099 

10  —  15 

.0189 

.972 

.985 

.783 

.0184 

.0186 

.0148 

15  —  20 

.0262 

.953 

.974 

.703 

.0250 

.0255 

.0184 

20  —  25 

.  0334 

.928 

.961 

.628 

.0310 

.0321 

.0210 

25  —  30 

.0403 

.898 

.947 

.  557 

.  0362 

.0382 

.0224 

30  —  35 

.0469 

.863 

.931 

.494 

.0405 

.0437 

.0232 

35  —  40 

.0531 

.824 

.915 

.440 

.0438 

.0486 

.0234 

40  —  45 

.0589 

.781 

.898 

.392 

.0460 

.  0529 

.0231 

45  —  50 

.0643 

.734 

.880 

.348 

.0472 

.0566 

.0224 

50  —  55 

.0692 

.684 

.862 

.310 

.0473 

.  0597 

.0215 

55  —  60 

.0736 

.633 

.844 

.275 

.0466 

.0621 

.0202 

60  —  65 

.0774 

.581 

.826 

.243 

.0450 

.0639 

.0188 

65  —  70 

.0806 

.531 

.808 

.213 

.0428 

.0651 

.0172 

70  —  75 

.0832 

.481 

.790 

.185 

.0400 

.0657 

.0154 

75  —  80 

.  0852 

.434 

.772 

.159 

.0370 

.0658 

.  0135 

80  —  85 

.0865 

.388 

.755 

.136 

.0336 

.0653 

.0118 

85  —  90 

.0872 

.342 

.738 

.'115 

.  0298 

.0644 

.0100 

90  —  95 

.0872 

.298 

.721 

.096 

.0260 

.0629 

.0084 

95  —100 

.0865 

.256 

.705 

.080 

.0221 

.0610 

.0069 

100  —105 

.0852 

.218 

.690 

.066 

.0186 

.0588 

.0056 

105  —110 

.0832 

.183 

.675 

.054 

.0152 

.  0562 

.0045 

110  —115 

.0806 

.150 

.661 

.044 

.0121 

.0533 

.0035 

115  —120 

.0774 

.120 

.648 

.035 

.0093 

.0502 

.0027 

120  —125 

.0736 

.094 

.636 

.028 

.0069 

.0468 

.0021 

125  —130 

.0692 

.073 

.625 

.021 

.0050 

.0433 

.0015 

130  —135 

.0643 

.055 

.614 

.016 

.0035 

.0395 

.0010 

135  —140 

.0589 

.040 

.605 

.012 

.0024 

.0356 

.0007 

140  —145 

.0531 

.028 

.596 

.010 

.0015 

.0316 

.0005 

145  —150 

.0469 

.019 

.589 

.008 

.0009 

.0276 

.0004 

150  —155 

.0403 

.015 

.582 

.006 

.0006 

.0235 

.0002 

155  —160 

.0334 

.012 

.577 

.005 

.0004 

.0193 

.0002 

160  —165 

.0262 

.010 

.572 

.005 

.0003 

.0150 

.0001 

165  —170 

.0189 

.008 

.569 

.004 

.0002 

.0108 

.0001 

170  —175 

.0114 

.006 

.567 

.004 

.0001 

.  0065 

.0001 

175  —180 

.0038 

.004 

.566 

.004 

.0000 

.0022 

.0000 

Sums 

2.0202 

0.7503 

1  .  4874 

0.3491 

LUNAR  AND  TERRESTRIAL  ALBEDOES 

DISCUSSION     OF    THE     OBSERVATIONS  1 

In  the  Astro  physical  Journal  for  April,  1916,  Professor  H.  N. 
Russell  has  discussed  some  recent  observations  of  the  writer  on 
the  earth-shine,  from  which  the  earth's  albedo  had  been  obtained 
indirectly.  The  observations  are  of  two  sorts — (i)  direct  visual 
comparisons  of  parts  of  the  moon  (lit  by  the  sun's  rays)  and  of 
other  parts  of  similar  quality,  lit  by  the  earth-shine,  with  the  light 
of  a  flame  seen  through  blue  glass;  and  (2)  comparisons  of  the 
relative  intensities  of  all  the  colors  in  the  spectrum  between 
violet  and  red  from  their  relative  photographic  effect  on  spectro- 
grams. 

The  earth-shine  spectrograms,2  along  with  similar  ones  of  the 
moon  and  of  the  sky,  were  measured  at  the  Westwood  Observ- 
atory by  means  of  the  comparator  originally  designed  for  quantita- 
tive measures  of  the  intensities  of  atmospheric  bands  on  the 
Lowell  Observatory  spectrograms  of  Mars  and  the  Moon.  With 
this  instrument  I  have  already  obtained  approximate  determina- 
tions of  the  amounts  of  the  water  vapor  and  oxygen  in  the  at- 
mosphere of  Mars,  and  I  am  now  engaged  in  measuring  the  in- 
tensities of  the  Fraunhofer  lines  in  the  solar  spectrum  with  an 
accuracy  which  has  not  been  approached  hitherto.  These  facts 
show  that  on  the  score  of  precision,  the  comparator  is  capable  of 
excellent  work,  though,  like  all  instruments  dependent  on  photo- 
graphy for  the  registration  of  intensities,  it  involves  the  complex- 
ities of  photographic  laws.  These  complexities,  however,  I  have 

1  The  essential  features  of  this  discussion  were  presented  before 
the   American   Astronomical    Society   at   its    Nineteenth    Meeting   at 
Swarthmore,  Pennsylvania,  August  31,  1916. 

2  The  spectrograms  were  made  for  me  through  the  kindness  of 
Dr.   Percival  Lowell  by  Dr.  V.  M.  Slipher  at  Flagstaff.     Several  of 
them  were  taken  under  exceptionally  favorable  atmospheric  conditions. 
See  Frank  W.  Very — "The  Photographic  Spectrography  of  the  Earth- 
Shine,"  Astronomische  Nuchrichten,  Nr.  4819-20,  Bd.  201,  s.  353-400, 
November,    1915;    also  "Atmospheric    Transmission,"   Science,   N.    S., 
Vol.  XLIV.,  No.  1127,  pages  168-171,  August  4,  1916. 

15 


16    LUNAR  AND  TERRESTRIAL  ALBEDOES 

endeavored  to  minimize,  and  I  have  in  large  part  succeeded  in 
eliminating  them  by  an  extensive  study  of  the  photographic  prob- 
lem for  each  wave-length  and  a  wide  range  of  exposures  on  more 
than  one  kind  of  plate.  Professor  Russell's  criticisms  of  my  work 
with  the  spectral  line  and  band  comparator  are  largely  founded  on 
misapprehensions.  He  has  taken  unwarrantable  liberties  with  my 
figures  and  by  so  doing  has  rejected  my  work  on  the  spectro- 
grams on  insufficient  grounds.  When  correctly  reduced,  the  two 
methods  give  results  which  are  in  good  agreement,  but  on  the 
whole,  those  from  the  spectrograms  are  the  more  reliable.1 

In  his  first  article,2  Russell  adopts  465,000  :  I  for  the  ratio 
of  sunlight  to  full-moon  light ;  and  on  page  184  of  the  April  Jour- 
nal he  expresses  preference  for  the  ratio  9,000  :  1 3  between  sun- 
light and  full-earth  light  on  the  moon.  According  to  this,  the  ratio 
of  full-earth  light  to  full-moon  light  is 

465,000   :  9,000  ==  51.7  :  i, 

1  The  visual  photometric  values  of  the  earth-shine  which  are  de- 
scribed in  my  paper  on  "The  Earth's  Albedo"   (Astronomische  Nach- 
richten,  Nr.  4696,  Bd.  196,  s.  269-290,  November,  1913)  were  obtained 
with  a  special  earth-shine  photometer  which  might  be  improved  in  the 
light  of  the  experience  gained  with  it.     Although  free  from  photo- 
graphic difficulties,  the  method  has  difficulties  of  its  own,  as  may  be 
recognized  from  the  elaborate  researches  which  were  required  in  estab- 
lishing the  constants  of  the  various  absorbent  pieces.     Owing  to  the 
faintness  of  the  earth-shine,  the  low  altitude  of  the  crescent  moon 
when  the  measures  have  to  be  made,  and  the  varying  transparency  of 
the  atmosphere,  there  are  further  difficulties  which  Professor  Russell 
generously  allows  for  in  his  criticism,  but  he  has  not  understood  some 
of  the  minor  details. 

It  must  be  remembered  that  the  spectrograms  were  made  with  an 
analyzing  spectroscope,  and  that  the  values  obtained  relate  to  the 
intrinsic  brightness  of  definite  regions  on  the  moon  where  the  reflect- 
ing quality  of  the  surface  is  far  from  uniform,  and  the  range  of 
luminous  values  with  the  phase  is  wide,  so  that  small  displacements  on 
the  surface  may  give  considerable  alteration  of  light.  The  observed 
differences  are  due  to  these  unavoidable  vicissitudes,  rather  than  to 
errors  of  observation;  but  such  differences  as  these  tend  to  average 
out  from  the  general  mean.  In  the  earth-shine  exposures,  the  slit 
was  placed  half  on  and  half  outside  the  dark  limb  of  the  moon  to  give 
the  sky  spectrum  needed  in  the  reductions. 

2  Astrophysical  Journal  for  March,  1916,  p.  125. 

3  On  page  194  (op.  cit.) ,  this  ratio  is  attributed  to  me,  but  I  have 
not  given  it,  and  prefer  the  ratio  10,000   :  1. 


LUNAR  AND  TERRESTRIAL  ALB  EDO  ES  17 

which,  since  the  angular  area  of  the  earth  as  seen  from  the  moon 
is  13.4  times  that  of  the  moon  seen  from  the  earth,  makes  the 
earth's  albedo  51.7/13.4  =  3.86  times  that  of  the  moon.  My 
own  determination  of  this  ratio  is  considerably  larger,  namely, 
4.8  :  i. 

The  ratio  of  sunlight  to  moonlight  is  not  easily  measured 
with  precision  on  account  of  the  wide  range  of  intensities  involved 
and  the  uncertainties  of  atmospheric  absorption.  Whether  we 
take  the  ratio  465,000  :  i,  difference  of  magnitude  =  14.17 
(Russell),  or  618,000  :  i,  difference  of  magnitude  =  14.48 
(Zollner),  or  even  a  value  as  large  as  Wollaston's  (801,000  :  i) 
we  shall  still  be  inside  the  actual  divergences  of  some  very  good 
observers.  A  critical  examination  of  sources  of  error  will  im- 
prove this  result  greatly. 

Zollner's  explanation  of  the  peculiar  efficacy  of  the  lunar 
mountains  in  emphasizing  the  peak  of  the  lunar  phase-curve  at 
the  full,  with  Searle's  emendation  which  notes  the  contribution 
to  the  same  effect  by  crevices  which  retain  the  sunshine  away 
from  observation  until  the  short  interval  when  the  rays  strike 
their  floors,  suffices  for  the  anomalies  of  this  phase-curve.  Wheth- 
er the  elevations  are  mountains,  as  is  usually  assumed,  or  innu- 
merable crystalline  facets,  that  is,  whether  the  roughness  is  on 
a  large  or  a  small  scale,  is  a  matter  of  indifference.  Zollner's 
diagrammatic  figure  is  well  characterized  by  Russell  as  "artifi- 
cial," but  the  fact  of  a  general  excessive  roughness  of  the  lunar 
surface  is  probable  enough  and  natural  enough,  even  though 
it  may  not  be  as  obvious  as  the  mountains  are  to  telescopic  vision. 
Nevertheless,  although  Zollner's  hypothetical  moon  behaves  in 
some  respects  like  the  actual  moon  near  the  time  of  full,  the  anal- 
ogy would  fail  if  pushed  to  its  limit,  especially  in  the  early 
crescent  phases,  and  the  title  "true"  which  was  used  by  Zollner 
to  designate  the  "albedo"  of  this  hypothetical  body  is  a  misnomer 
when  applied  to  the  actual  moon.  Russell's  criticism  by  -ques- 
tioning the  foundation  of  an  almost  unanimous  acceptance  of 
Zollner's  value  and  terminology,  performs  a  much  needed  use. 
In  this  he  has  followed  Guthnick  more  or  less.  Miiller,  also,  had 
previously  characterized  Zollner's  value  of  the  mean  (52°)  slope 
of  the  lunar  upheavals  as  "illusory." 

In  getting  the  lunar  albedo,  Zollner  allowed  for  the  rotundity 


18    LUNAR  AND  TERRESTRIAL  ALBEDOES 

of  the  moon,  and  by  an  unfortunate  misapplication  of  Lambert's 
law  of  reflection  from  a  uniformly  diffusing  sphere1  found  that 
the  full  moon  should  reflect 


P        ( 3/2  )X  48,980       73,470 

of  sunlight,  if  it  were  such  a  sphere.  The  corresponding 
value  of  the  albedo  he  called  the  "apparent  albedo"  (scheinbare 
Albedo) — a  term  which  is  misleading,  since  it  implies  that 
the  moon's  reflection  which  is  actually  observed  is  this  quantity 
given  by  computation,  and  that  the  reflection  up  to  this  point 
follows  Lambert's  law,  which  is  not  true. 

The  total  radiation  of  the  moon,  including  both  reflected  and 
emitted  rays,  does  appear  to  follow  pretty  nearly  a  sequence 
which  can  be  derived  from  Lambert's  law,  multiplied  by  the 
factor  2/3,  a  number  which  seems  to  turn  up  on  every  hand  in 
this  research.  This  may  be  seen  from  the  phase-curve  given  in 
my  "Prize  Essay  on  the  Distribution  of  the  Moon's  Heat  and 
its  Variation  with  the  Phase,"  where  I  found 
At  first  quarter,  total  radiation  =  17.7%  of  radiation  from 

full  moon. 
At    last    quarter,    total    radiation  =    24.8%   of   radiation    from 

full  moon. 
Mean  (quadrature)  total  radiation  =  21.25%  °f   radiation   from 

full  moon. 
Lambert's  law,  multiplied  by  2/3  =  21.22%   of   radiation    from 

full  moon. 

Here  the  radiation  which  is  measured,  is  made  up  of  two 
parts — one  which  is  wholly  reflected,  and  the  other  an  emission 
from  a  heated  surface  whose  temperature  varies  from  a  maxi- 
mum at  the  center  of  the  disk  to  a  minimum  at  the  limb,  while 
the  surfaces  of  equal  temperature  are  concentric  zones.  The 
luminous  reflection,  at  any  rate,  follows  the  opposite  law  and  is 
greater  at  the  limb,  and  probably  the  non-luminous  rays  are 
reflected  in  the  same  way.  The  smaller  total  radiation  at  first 
quarter  is  of  course  due  to  the  fact  that  the  moon  is  getting  hotter 
and  the  energy  is  being  expended  in  modifying  the  subsurface 

1  The  law  was  designed  to  give  the  spherical  albedo.  What  was 
needed  here  was  simply  the  geometrical  albedo. 


LUNAR  AND  TERRESTRIAL  ALB  EDGES  19 

thermal  gradient,  while  the  heat  thus  retained  is  given  out  again 
in  the  lunar  afternoon,  or  at  last  quarter.  The  point  which  I 
wish  to  make  clear  is  that,  as  far  as  the  reflection  of  the  moon's 
lit/ lit  is  concerned,  the  introduction  of  Lambert's  law  at  this 
or  any  other  stage  of  the  computation  was  a  mistake. 

The  value  "p  =  0.1195,"  which  Zollner  calls  "die  scheinbare 
Albedo  des  Mondes,"  is  a  hypothetical  value  which  is  not  imme- 
diately given  by  observation,  but  is  obtained  by  restricting  the 
definition  of  albedo  to  diffuse  reflection  from  a  nonexistent  smooth 
sphere  on  the  supposition  that  a  factor  2/3  must  be  introduced. 
I  will  return  to  this  later,  but  will  note  here  that  the  moon  does 
not  reflect  much  after  the  approved  fashion  of  a  sphere,  but  acts, 
to  all  appearance,  more  like  a  flat  surface,  or  even  like  one  a 
little  dished  at  the  margin ;  for  whereas  a  diffusive  sphere  should 
send  out  light  from  the  marginal  zones  at  full  into  a  rear  hem- 
isphere, whereby  these  zones  as  seen  from  the  front  should  be 
considerably  fainter  than  the  center,  it  is  found,  on  the  contrary, 
that  the  limb  in  the  actual  moon  is  in  fact  the  brighter  of  the  two. 
Both  front  and  rear  reflections  from  the  limb  are  exceptionally 
large,  so  that  these  portions  of  the  lunar  surface  are  but  little 
heated  by  the  sun's  rays. 

The  arguments  by  which  Zollner  persuaded  himself  that  he 
had  arrived  at  the  "true"  value  of  the  moon's  albedo  from  his 
"apparent"  albedo  are  somewhat  involved.  One  of  the  first  needs 
is  that  our  procedures  and  definitions  may  be  -clarified  and  sim- 
plified. Let  us  begin  by  considering  the  Lambert  law. 

On  page  176  of  his  second  article,  Professor  Russell  says 
correctly  in  speaking  of  the  reflective  function  of  the  phase-angle, 
cp(a)  :  "The  whole  amount  of  light  reflected  by  the  planet  to  the 
celestial  sphere  will  be  proportional  to 


f 

c/    t 


cp(a)  sin  a  da. 


If  it  shone  in  all  directions  with  the  brightness  of 
the  full  phase,  the  emitted  light  would  be  2.0  on 
the  same  scale."  But  this  being  so,  the  factor  q  which 
transforms  from  the  geometrical  albedo  at  opposition  to  the 
spherical  albedo  should  be  the  integral  just  given,  and  not  twice 


20     LUNAR  AND  TERRESTRIAL  ALBEDOES 

that  quantity,  which  is  Russell's  equation  (7),  because  the  value 
of  the  integral  alone  without  the  coefficient  2  is  2.0,  if  <p(a)  =  i 
everywhere.  For  Lambert's  law, 

f  * 

(6)  q  =  I      <p(a)    sin  a  e?a  — 0.75, 


=  f 


and  this  is  the  Lambert  factor  for  spherical  albedo,  or 
it  is  the  spherical  albedo  if  the  geometrical  albedo  at 
full  phase  is  unity,  instead  of  #=1.5  as  given  by  Russell. 
Similarly,  for  the  lunar  phase-curve,  q=O-3$,  which  is 
to  be  multiplied  into  the  observed  geometrical  albedo  from 
the  full  moon  to  give  the  lunar  spherical  albedo.  Since  the  values 
of  q  have  been  taken  two  times  too  large,  all  of  the  numbers  in 
Russell's  Table  I.,  op.  cit.,  page  179,  should  be  divided  by  two.  On 
the  other  hand,  in  getting  q  for  the  Lommel-Seeliger  law,  Russell 
has  inconsistently  dropped  the  factor  2,  which  would  make  his 
value  the  same  as  mine,  were  it  not  that  there  is  a  further  in- 
accuracy in  the  integration  by  which  he  gets  "(7=1.6366."  His 
equation  (7)  should  give  ^  =  3.0. 

Lambert's  formula  for  spherical  albedo, 

L  =   (I/JD)    [sin  a  —  a  cos    a] -)- cos  a, 

where  a  is  the  moon's  elongation,  or  phase-angle  from  conjunc- 
tion, or  as  Russell  prefers  to  put  it,  employing  phase-angles  from 
opposition',  (in  which  respect  I  shall  follow  his  procedure) 

<p(a)   =    (i/ft)    [sin  a-f-  (:t  —  a)  cos  a], 

gives  the  light  at  quadrature,  L90  =  i/jt  =  0.318,  when  the  light 
at  the  full  phase  is  unity.  For  the  emission  of  its  own  radiation 
combined  with  the  reflection,  the  radiant  observation  already 
quoted  would  seem  to  favor  the  value,  RQO  =  i/(  1.511)  =  0.212. 
Here,  however,  we  must  note  that,  though  the  emission  may  be 
independent  of  its  direction,  the  distribution  of  temperature  is 
not  uniform,  and  the  result  is  a  complex  of  two  different 
functions  of  a. 

Under  these  circumstances  we  can  attach  little  significance 
to  this  special  value.  It  is  difficult  to  see  how  a  particular  numer- 
ical factor  can  survive  this  double  vicissitude.  If  the  factor  2/3 
occurs  undisguised  in  both  functions,  it  will  become  4/9 
in  their  product;  if  in  opposite  senses,  it  will  cancel  out; 


LUNAR  AND  TERRESTRIAL  ALBEDOES  21 

and  if  the   factor  is   found  in  one  case,  but  a  different  one  is 
substituted  for  it  in  the  other,  it  could  not  be  so  easily  recognized. 
By  Lambert's  law,  at  the  full,  L0  =  jtX  ^-90  —  3-T4  -^90- 
For  the  planet  Venus,  reflection  =  L0  =  (17.283)  L90  =  3-53^90- 
Lunar  emitted  and  reflected  radiation  =  R0  =  (3/2)  JtX  -^90  — 


90. 


Lunar  reflected  light  =  L0  =  (i/.ios)  Lao  =  9-52L9 

The  factors  connecting  the  observed  light  at  full  phase 
with  the  light  at  quadrature  are  evidently  empirical.  Those 
connecting  spherical  and  geometrical  albedoes  are  probably 
equally  empirical.  The  appearance  of  the  factor  (3/2)  Ji  in  my 
lunar  radiation  observation,  to  which  I  have  called  attention,  is 
rather  striking,  yet  it  is  probably  no  better  than  a  coincidence. 
Zollner  must  have  been  under  a  great  misapprehension  when  he 
attempted  to  introduce  the  factor  3/2  into  the  discussion  of  his 
lunar  observations,  for  it  does  not  fit  the  facts. 

There  is,  it  is  true,  a  universal  usage  for  which  the  factor 
3/2  is  appropriate,  namely  :  The  reflection  from  a  sphere  of  any 
ordinarily  diffusive  material  is  2/3  of  that  returned  with  per- 
pendicular incidence  and  reflection  from  a  plane  surface  of  the 
same  substance,  and  in  comparing  the  reflection  of  the  entire 
spherical  body  of  a  planet  with  that  from  a  plane  surface  of 
some  terrestrial  substance,  it  would  be  appropriate,  provided  we 
could  be  sure  that  the  light  is  diffusively  reflected,  to  multiply 
the  reflection  from  the  sphere  by  3/2  to  put  it  on  terms  of 
equality  with  the  recognized  reflective  power,  or  specific  reflect- 
ivity, of  the  given  terrestrial  material  in  the  form  of  a  flat  surface 
when  this  is  viewed  normally.  If  the  flat  surface  reflects  the 
light  at  an  angle  of  45°,  its  reflection  must  .  be  multiplied  by 
cos  45°  =  0.707,  and  in  this  case  the  two  reflectors  are  already  on 
terms  of  approximate  equality.  It  does  not  appear  that  the  factor 
3/2  was  introduced  by  Zollner  with  any  such  end  as  this  in  view, 
but  simply  because  it  occurs  in  the  Lambert  formula.  As  applied 
by  Zollner  in  his  lunar  theory,  this  use  of  the  factor  3/2  has  been 
a  stumbling  block  in  the  path  of  subsequent  research  on  the  subject. 
The  whole  of  this  cloudy  lucubration  should  be  swept  aside. 
Zollner's  original  lunar  observations  are  among  the  best  that 
have  ever  been  made,  and  they  deserve  to  be  rescued  from  the 
scandalous  treatment  of  his  theoretical  discussion. 


22     LUNAR  AND  TERRESTRIAL  ALBEDOES 

At  the  start,  Zollner  has  evidently  adopted  the  incorrect  idea 
that  the  fraction  of  light  from  the  sun  received  upon  the  moon's 
surface  and  which  has  to  be  considered,  is  the  fraction  of  the 
total  luminous  output  of  the  sun  to  the  entire  sphere,  and  he  thus 
gets  for  the  denominator  the  number  48,980.  He  sees  that  this 
number  is  too  small  for  his  observations  and  makes  the  hypothesis 
that  it  must  be  multiplied  by  3/2,  giving  73,470,  to  which  he 
assigned  the  symbol  p.  But  this  is  in  turn  too  small,  and  he 
introduces  the  further  hypothesis  of  the  lunar  mountains  with 
slopes  of  52°,  getting  a  new  factor  x,  and  x/»  =  107,300,  with 
which  his  value  of  the  so-called  "true"  albedo,  0.174,  was  obtained 
by  comparing  it  with  the  observed  ratio  of  sunlight  to  moonlight. 
If  he  had  started  out  with  the  correct  conception  that  the  moon 
receives  a  certain  fraction  of  the  light  emitted  by  the  half  sphere 
of  the  sun  which  is  turned  toward  the  planet,  he  would  have 
obtained  at  once  the  number  97,960  (slightly  different  from  the 
one  which  I  have  used,  because  we  have  adopted  slightly  different 
values  of  the  moon's  semidiameter)  and  he  would  have 
obtained  at  once  for  the  geometrical  albedo  of  the  moon, 
97,960/618,000  =  0.1585. 

In  short,  Zollner  started  with  a  wrong  number,  multiplied 
this  by  3/2  and  then  by  nearly  another  3/2,  or  in  all  by  nearly 
2%,  instead  of  by  2  exactly,  as  he  should  have  done,  and  thus 
by  a  threefold  error  he  reached  a  result  which  was  nearly  right, 
but  solely  by  accident. 

Russell  starts  with  the  same  erroneous  conception  that  the 
sun's  complete  spherical  emission  should  be  considered,  but  im- 
mediately abandons  it  (though  without  noting  the  fact)  for 
another,  not  necessarily  incorrect,  but  different  from  mine,  since 
he  does  not  introduce  4  into  the  numerator  of  his  expression  for 
p,  nor  yet  the  number  2  which  would  give  what  I  call  the  geomet- 
rical albedo,  but  multiplies  by  i,  whence  his  p  is  one  half  of 
the  geometrical  albedo,  given  by  eye  observation,  and  represents 
surface  illumination  as  I  have  shown  in  the  Introduction. 
He  then  makes  the  reverse  change  by  introducing  2  into 
his  value  of  q,  outside  the  integral,  where  it  does 
not  belong  if  by  q  is  meant  the  spherical  factor,  as  in  my 
equation  (6),  so  that  his  q  is  two  times  mine  and  through  the 


LUNAR  AND  TERRESTRIAL  ALBEDOES     23 

cancellation  of  these  opposite  transformations  we  finally  reach 
similar  results  for  the  spherical  albedo. 

The  geometrical  albedo,  Av  which,  like  Russell's  p,  "depends 
only  on  the  geometrical  and  photometric  relations  of  the  planet 
as  observed  at  the  full  phase"  1  is  correctly  given  by  equation  (7), 

2A/(1  sin2  S 

(7)  A,  =  ~^-     .  2       , 

sin    ,y    sin    o 

where  S  is  the  apparent  semi-diameter  of  the  sun  as  seen  from 
the  earth,  ,y0  and  <TO  are  the  semi-diameters  of  the  sun  as  seen 
from  the  planet  and  of  the  planet  as  seen  from  the  earth  at  the 
time  of  opposition,  and  M0  is  the  ratio  of  the  light  received  from 
the  planet  at  mean  opposition,  to  the  light  of  the  sun  as  observed 
from  the  earth.  This  equation  is  the  same  as  the  middle  one 
of  Miiller's  (i4)2  and  is  a  special  case  derived  from  the  Lommel- 
Seeliger  theory,  which,  although  it  is  a  theory  of  spherical 
reflection,  gives  the  geometrical  albedo  at  this  particular  point,3 
which  Russell's  equation  (5)  does  not  do. 

Professor  Russell  says:  "Let  r  be  the  mean  radius  of  the 
planet's  disk,  and  R  its  distance  from  the  sun,  and  M0  be  the  ratio 
of  the  apparent  brightness  of  the  planet  at  the  full  phase,  and  at 
distance  A  from  the  earth,  to  that  of  the  sun  at  unit  distance. 
The  fraction  of  the  sun's  whole  radiation  which  the  planet  in- 
tercepts is  r2/4R2."  *  This  is  all  true,  but  it  is  not  what  we  want  to 
know.  The  fraction  of  the  sun's  light  emitted  by  the  solar  hem- 
isphere which  is  visible  from  the  planet,  and  which  the  planet 
intercepts,  is  nr2/2nR2  =  rz/2.R2;  and  M0  being  the  ratio  of  the 
observed  planetary  light  at  full  phase  to  sunlight  and  A  the 
distance  from  the  earth  at  the  time  of  observation,  so  that  if  the 
other  things  remain  the  same,  Af0A2  =  const.,  an  alternative  ex- 
pression for  the  geometrical  albedo  is 


Tt  is  not  necessary  in  this  problem  of  the  reflection  of  the  sun's 

1  H.  N.  Russell,  in  Astrophysical  Journal  for  April,  1916,  p.  177. 

2  Photometric  der  Gestirne,  p.  65. 

3  The  geometrical   albedo   is   in  fact  the   same   as   the   spherical 
albedo  on  the  assumption  that  the  spherical  factor  is  unity  (q  =  1). 

4  Astrophysical  Journal,  April,  1916,  p.  176. 


24  LUNAR  AND  TERRESTRIAL  ALBEDOES 

rays  to  consider  the  sun's  invisible  hemisphere.  The  reflection 
would  be  the  same  if  this  were  dark.  For  the  particular  problem 
under  discussion,  the  other  side  of  the  sun  is  as  if  it  were  non- 
existent, and  it  has  nothing  to  do  with  the  question.  That  there 
may  be  no  misunderstanding,  I  repeat  that  Russell's  final  expression 
for  what  he  denotes  under  the  symbol  p,  and  which  I  call  surface 
illumination,  must  be  multiplied  by  2  in  order  to  obtain  the 
quantity  A2  or  the  geometrical  albedo;  and  therefore  his  factor 
"q"  which  reduces  to  spherical  albedo  is  to  be  divided  by  2,  so 
that,  apart  from  all  other  considerations,  that  is,  granting  the 
reliability  of  the  original  data,  there  should  be  no  difference  be- 
tween us  in  respect  to  the  spherical  albedoes  ("A"  of  Russell's 
Table  V),  provided  we  could  agree  in  regard  to  the  best  phase- 
law  to  be  adopted.1 

The  quantity  "p"  is  denned  in  two  ways  in  Russell's  paper : 
"The  factor  p  may  also  be  defined  as  the  ratio  of  the  actual 
brightness  of  the  planet  at  the  full  phase  to  that  of  a  self-luminous 
body  of  the  same  size  and  position,  which  radiates  as  much  light 
from  each  unit  of  its  surface  as  the  planet  receives  from  the  sun 
under  normal  illumination"  (Op.  cit.,  pp.  177-178). 

By  this  definition,  as  interpreted  by  Russell,  the  quantity  p  is 
proportional  to  Miiller's  Af0,  since  all  of  the  planetary  values  in 
Russell's  equation  have  been  reduced  to  unit  distance,  and  Afn 
is  a  ratio  to  sunshine  at  unit  distance.  But  from  (7) 

M0  =  J-^oX  const., 

or  Russell's  p,  like  Miiller's  A/0,  is  proportional  to  the  half  of  the 
geometrical  albedo. 

The  writer  would  interpret  the  definition  itself  differently, 
because  a  planet  of  radius  r  and  distance  from  the  sun  R,  will 
receive  from  the  sun,  if  L  is  the  sun's  total  spherical  emission  of 
light,  the  light-quantity  L/4nR2  on  each  unit  of  normally  ex- 
posed surface.  But  the  planet  "receives"  light  from  the  sun  on 
only  one  half  of  the  planetary  surface,  and  hence,  if  it  radiates 
from  "each  unit  of  its  surface"  self-luminously  (or  what  amounts 
.  to  the  same  thing,  if  the  light  falling  normally  on  a  plane  surface 
equal  to  the  planet's  section,  is  wholly  emitted  by  a  hemispherical 
surface  of  that  planet)  the  emitted  light  is  given  out  through 

1  This  will  be  considered  in  a  separate  paper. 


LUNAR  AND  TERRESTRIAL  ALB  EDO  ES  25 

a  surface  twice  as  great  as  the  receiving  surface,  and  should  be 
on  the  average  intrinsically  half  as  intense  as  the  received  light. 
If  the  intrinsic  brightness  of  the  emitted  light  is 


the  total  light  emitted  by  the  sphere  is 

L    r~ 


L'  =  4JL/V2  = 

But  this  does  not  represent  the  real  conditions,  because  the  defi- 
nition itself  is  faulty,  since  the  planet  does  not  receive  light 
"under  normal  illumination"  on  every  part  of  its  exposed  hem- 
isphere. If,  therefore,  we  make  my  parenthetical  substitution, 
and  omit  the  stipulation  that  each  unit  of  surface  shall  radiate 
as  much  light  as  it  receives  "under  normal  illumination,"  the 
total  light  emitted  by  the  hemisphere  is 

I     r2 

L'  =  znJr*  =  -     -, 
4    K- 

and  we  have  returned  to  the  previous  proposition  relating 
to  the  fraction  of  the  sun's  whole  spherical  emission  which 
the  planet  intercepts.  This,  as  I  have  said,  does  not 
concern  us  in  the  problem  of  reflection,  where  the  planet 
reflects  light  solely  from  the  visible  hemisphere  of  the  sun, 
and  the  invisible  solar  hemisphere  with  its  emission  need  not  be 
considered,  since  none  of  its  light  reaches  the  planet. 

\Ve   have,   therefore,   for  the   reflection-ratio,   if  all  of  the 
sunlight  is  reflected, 

.  L  L'         r- 

L/~  =  2  T-  = 


2  L  2R2 

and  the  geometrical  albedo  is  the  ratio  of  the  observed 
reflection  to  this  value ;  whence  Russell's  first  definition, 
if  modified  so  as  to  bring  it  into  accord  with  the  actual 
conditions,  agrees  with  my  definition  of  the  geometrical  albedo. 

On  the  other  hand,  Russell's  second  definition  can  not  be 
thus  reconciled.  It  reads  as  follows : 

"The  factor  p  may  be  defined  verbally  as  the  ratio  of  the 
observed  brightness  of  the  planet  at  full  phase  to  that  of  a  flat 
disk  of  the  same  size  and  in  the  same  position,  illuminated  and 


26     LUNAR  AND  TERRESTRIAL  ALBEDOES 

viewed  normally,  and  reflecting  all  the  incident  light  in  accordance 
with  Lambert's  law"  (op.  cit.,  p.  188). 

If  the  sphere  shines  "in  all  directions  with  the  brightness 
of  the  full  phase"  (op.  cit.,  p.  176),  the  quantity  which  I  call  q 
(eq.  6)  is 

q  —  fp(a)    sin  a  da  =  2.0. 


y. 

*J      0 


But  if  the  sphere  shines  according  to  Lambert's  law,  the  value 
of  the  integral  is  3/4.  If,  however,  the  further  proviso  be  made 
that  p  is  "the  ratio  of  the  observed  brightness  of  the  planet  at 
full  phase  to  that  of  a  flat  disk,"  since  the  ratio  of  reflection  from 
a  sphere  illuminated  and  viewed  from  the  front  (or  that  reflection 
corresponding  to  its  geometrical  albedo),  is  to  the  reflection  from  a 
flat  disk,  illuminated  and  viewed  normally,  as  2/3  :  i,  the  3/4 
must  be  multiplied  by  2/3  giving  1/2  as  the  maximum  "flat-disk" 
value  of  the  spherical  albedo.  The  same  fraction  expresses  the 
relation  between  the  intregral  computed  for  cp(ct)  as  given  by 
Lambert's  law  (q±)  and  by  the  Lommel-Seeliger  law  (q2), 
namely, 


Since,  however,  the  total  reflection,  or  spherical  albedo,  must  be 
the  same  and  equal  to  the  incident  light  on  either  hypothesis 
•with  complete  and  diffusive  reflection,  it  follows  that  whatever 
differences  result  from  these  hypotheses  must  fall  upon  q  and  p 
equably  and  oppositely  in  order  that  their  product,  A  =  qp,  may 
remain  the  same  for  the  given  planet.  That  is  to  say,  whatever 
variations  there  may  be  in  the  distribution  of  light  to  different 
zones  by  reflection  according  to  the  rival  theories,  the  sum  total, 
or  spherical  albedo,  must  be  the  same  for  either  if  the  reflection 
is  complete.  Hence  if  q  is  obtained  by  Lambert's  law,  p  must  be 
twice  as  large  as  it  would  be  if  q  were  given  by  the  Lommel- 
Seeliger  law.  There  appears,  therefore,  to  be  an  inconsistency  in 
Russell's  second  definition,  and  his  "p"  matches  the  condition 
demanded  by  the  Lommel-Seeliger  law,  while  my  doubled  value, 
though  given  as  a  particular  case  of  the  Lommel-Seeliger  formula, 
has  to  be  combined  with  the  Lambert  value  of  q,  if  A  =  qp  is  to 
be  kept  constant.  We  thus  reach  the  curious  dilemma  that 


LUNAR  AND  TERRESTRIAL  ALB  EDO  ES  27 

neither  of  these  two  rival  theories  can  dispense  with  the  other. 
As  if  to  enforce  this  point,  the  actual  phase-curve  of  Venus  fol- 
lows a  mixture  of  the  two  laws.  It  becomes  exceedingly  difficult 
to  "mind  your  p's  and  q's,"  under  these  circumstances. 

If  we  knew  the  mean  temperature  of  a  planet  for  all  latitudes 
from  the  equator  to  the  poles,  we  should  no  doubt  find  some 
relation  between  the  thermal  quantity  corresponding  to  this  tem- 
perature and  the  spherical  albedo  of  the*planet.  In  general,  we 
may  anticipate  that  the  greater  the  spherical  albedo  is,  the  less  will 
be  the  heat,  but  not  necessarily  in  any  exact  proportionality,  be- 
cause the  blanketing  action  of  the  planet's  atmosphere  is  the 
principal  factor  in  the  retention  of  any  heat  which  the  surface 
may  receive.  Many  geologists  believe  that  a  continually  cloudy 
atmosphere,  which  would  certainly  reflect  most  of  the  sun's  rays, 
but  would  retain  terrestrial  heat,  was  largely  responsible  for  the 
growth  of  a  tropically  luxuriant  vegetation  within  the  Arctic 
circle  in  past  ages.  Morever,  some  highly  important  climatic 
properties,  such  as  the  melting  of  snow  in  high  latitudes  and  the 
possibility  or  nonpossibility  of  a  permanent  fee-cap,  depend  on  the 
maximum  summer  temperature,  rather  than  on  the  mean  tem- 
perature. The  maximum  temperature  at  the  sub-solar  point  is 
somewhat  intimately  related  to  the  reflection  of  total  radiation 
spherically,  and  to  some  extent  the  latter  follows  the  luminous 
albedo.  The  winter  snows  of  Mars  reach  nearly  as  low  a  latitude 
as  on  earth,  but  no  lower.  The  feebler  sunshine  of  Mars  is  better 
conserved  because  it  suffers  a  smaller  reflective  loss  in  passing 
through  a  clearer  and  rarer  atmosphere.  In  fact,  the  albedo  of 
Mars  is  so  much  smaller  than  the  earth's  that  Mars  would  have 
the  hotter  climate  of  the  two,  in  spite  of  greater  distance  from  the 
sun,  were  it  not  that  a  rare  atmosphere  permits  an  easier  escape 
of  Martian  surface  radiation.  On  a  planet  with  hardly  any  air, 
but  having  a  long  period  of  insolation  and  approximation  to  a 
steady  state  of  thermal  equilibrium,  the  sub-solar  effect  of  the 
sun's  rays  must  be  nearly  equal  to  the  solar  constant  multiplied  by 
one  minus  the  spherical  reflection  of  solar  rays  of  every  wave- 
length (i — ^!(t)).  Thus,  for  the  moon,  the  reflection  of  total 
radiation  in  connection  with  the  temperature  (both  of  which  are 
measurable)  has  a  bearing  on  the  problem  of  the  solar  constant, 


28  LUNAR  AND  TERRESTRIAL  ALBEDOES 

although  it  may  not  be  possible  to  utilize  the  information  fully  for 
lack  of  other  data. 

If  a  completely  and  uniformly  diffusive  reflecting  sphere  of 
indefinitely  great  radius  be  drawn  about  the  sun  as  a  center  and 
including  the  earth,  and  if  the  central  radius  of  the  segment  of 
this  sphere  in  view  from  the  night  side  of  the  earth  be  in  the 
prolongation  of  the  line  from  the  sun  to  the  earth,  or  what 
amounts  to  the  same  thing,  if  we  imagine  an  ideal  night  sky  to  be 
"packed"  with  perfectly  reflecting  full  moons,  such  a  segment  of 
the  sphere  (embraced  in  a  hemisphere  as  viewed  from  the  earth), 
or  such  a  sky,  should  reproduce  sunlight  at  the  earth's  distance. 
The  moon  sends  us  1/98,317  part  of  the  light  reflected  from  such 
a  hemisphere  to  a  point,1  and  the  full  moon,  which  may  be  likened 
to  a  circular  disk  of  15*32. "7  radius  cut  out  from  this  surface, 
should  send  us  that  fraction  of  sunlight,  if  it  were  not  that  it  does 
not  reflect  in  that  way,  but  absorbs  all  but  a  small  part  of  the  com- 
bined luminous  and  nonluminous  radiation  received  from  the  sun. 
Since,  however,  when  equilibrium  is  attained,  the  combined  emis- 
sion and  diffuse  reflection  of  rays  of  every  wave-length  must  equal 
the  total  of  solar  radiation  received  (unless  there  is  some  excep- 
tional specular  reflection  in  a  particular  direction,  of  which  there 
is  no  evidence,  and  except  for  a  slight  retention  of  heat  to  be 
radiated  away  during  the  night)  a  measurement  of  the  heat 
received  from  the  total  radiations  of  sun  and. moon,  respectively, 
should  approximate  to  this  ratio.  Such  a  measurement  was  made 
by  Director  Langley  and  myself  at  the  Allegheny  Observatory 
from  which  the  fraction  1/96,509  was  obtained,  which  seems  to 
be  in  sufficiently  close  agreement  with  the  theory. 

Combining  the  above-named  theoretical  value  with  the  observed 

ratio    of    at     full    moon,     we    have    the     following 

moonlight 

Values  of  the  geometrical  lunar  albedo : 
By  Zollner's  observation,  A2=  -  =0.159, 

1  Namely,  light  from  surface  of  hemisphere    :   light  from  lunar 

i 

Q  Ot  T^  °      ^  O 

disk  =  -.--•  -  =  __-_£_——-  x      1.00515  =  98,317  :  1. 

o,r>2    rr  gmJ   g  gm.>    Jg'  32". 7        A 


LUNAR  AND  TERRESTRIAL  ALBEDOES  29 

With  Russell's  adopted  ratio,  A2=    9  '3*7      =o.2ii,1 


08 

With   Miiller's  adopted   ratio,  ^2—-  =0.173. 

5^9'Soc* 

Mean  A,  =  0.181. 

Miiller  points  out  that  numerical  values  will  differ  according 
to  the  definitions  of  albedo  of  which  there  are  several.  Of  three 
theories  given  in  his  book  with  much  detail,  neither  one  is  even 
remotely  applicable  to  the  moon,  except  in  so  far  as  the  particular 
values  by  the  Lommel-Seeliger  and  Euler  theories  for  full  phase 
do  coincide  with  the  geometrical  albedo,  on  account  of  the  afore- 
said identity  of  the  equations  for  this  special  case.  The  albedoes 
"by  Seeliger's  definition"  which  are  set  down  by  Miiller  are  obtain- 
ed on  the  limiting  assumption  that  the  coefficients  of  absorption 
of  incoming  and  outgoing  rays  have  the  ratio  X=i,  which  makes 
the  coefficient  in  the  Seeliger  formula  for  A2  =  2,  or  the  same  as  in 
the  formula  for  geometrical  albedo.  Otherwise,  if  X  differs  from 
unity,  we  have 

A  =  i,  numerical  coefficient  =  2.0000 
A  =  2,  numerical  coefficient  =  1.8924 
>.  =  3,  numerical  coefficient  =  1.8679 
X  =  4,  numerical  coefficient  =  1.8628 
A,  =  5,  numerical  coefficient  —  1.8639 
A,  =  6,  numerical  coefficient  =  1.8673 
A.  =  10,  numerical  coefficient  =  1.8846 

The  albedoes  "by  Lambert's  definition"  are  spherical  albedoes  de- 
rived from  the  geometrical  albedoes  by  applying  the  factor 
q  =  0.75,  which  is  obtained  by  integration  of  the  Lambert  phase- 
curve.  Muller  leaves  the  reader  to  choose  for  himself  between 
these  values  of  the  moon's  albedo: 

"A1  =  0.129   (by  Lambert's  definition), 
^2  =  0.172  (by  Seeliger's  definition)."2 

The  use  of  the  Lambert  theory  and  of  the  constant  factor  0.75  in 
passing  from  A2  to  Alt  prevents  the  values  of  A1  in  Miiller's  book 
from  being  regarded  as  spherical  albedoes,  except  in  those  cases 
where  Lambert's  law  may  possibly  be  followed  approximately. 

1  Russell  himself,  as  already  noted,  divides  this  by  2  to  get  his 
"p,"  obtaining  p  =  0.105,  and  for  Zollner's  value,  p  =  0.08. 

2  Photometric  der  Gestirne,  p.  343. 


30     LUNAR  AND  TERRESTRIAL  ALBEDOES 

With  my  value  of  the  earth :  moon  ratio  and  Miiller's  geomet- 
rical  albedo   of    the   moon,    the    earth's   geometrical    albedo    is 

AC2  =  4.8  X  o-1?2  =  0-826. 

The  ratio  4.8   :  I  applies  only  to  the  geometrical  albedoes.     The 
spherical  albedoes  adopted  by  Russell,  namely, 

"A"  =  Ami  =  0.073  f°r  tne  moon, 
"A".=  Aei  =  0.45  for  the  earth, 

have  the  larger  ratio  Aei    :  Ami  =  6.16    :   i,  and  I  shall  show 
presently  that  this  ratio  ought  to  be  still  further  increased.     On 
the  other  hand  Russell's  values  of  p  which  are  proportional  to 
geometrical  albedoes  have  a  ratio  smaller  than  mine,  namely : 
^(e)    .  ^(m)  __  Aez  .  Am2  ==  3.86   :  i, 

and  one  which  does  not  agree  with  his  adopted  ratio  of  sunlight 
to  moonlight.  A  revision  on  this  account  is  certainly  required. 
Russell's  lunar  value,  "/>  =  0.105,"  if  ^  represents  the 
"reflecting  power"  of  the  lunar  surface,1  would  require 
that  the  moon  should  be  composed  of  something  almost  as  dark 
as  dark  grey  slate,  or  nearly  like  trachyte  lava,  o.io,  according 
to  the  figures  which  he  quotes  from  Wilsing  and  Scheiner.  But 
excluding  the  very  -brightest  and  darkest  spots  which  are  of  rela- 
tively small  area,  there  are  extensive  dark  regions  on  the  moon 
whose  average  total-radiation  reflection  (bolometrically  deter- 
mined by  measuring  the  transmission  of  lunar  radiation  through 
a  glass  plate  which  cuts  off  practically  all  of  the  emitted  rays  and 
distinguishes  between  these  and  the  reflected  ones)  is  from  10  to 
12  per  cent.,  while  that  of  correspondingly  situated  bright  regions 
(similarly  determined)  is  from  20  to  25  per  cent.  Since  the 
moon's  surface  is  about  equally  divided  between  such  "dark" 
and  "bright"  areas,  a  mean  total-radiation  reflection  of  0.15  to 
0.185  (average  =  0.168)  is  indicated  by  my  bolometric  measures 
which  form  a  useful  check  on  my  photometric  results.2 

1  Russell  says  (op.  cit.,  p.  192)  :  "Wilsing  and  Scheiner  have  de- 
termined the  reflecting  power  of  many  ordinary  rocks,  using  an  ap- 
proximately flat,  rough,  natural  surface  normal  to  the  incident  and 
reflected  rays.     Their  formula  of  reduction  gives  exactly  the  quantity 
which  has  been  designated  by  p."    To  the  writer,  it  looks  as  if  p,  the 
planetary  illuminating  power,  should  be  multiplied  by  3/2  before  mak- 
ing this  comparison. 

2  Some  samples  of  these  are  to  be  found  in  my  "Photometry  of  a 
Lunar  Eclipse,"  Astrophysical  Journal,  November,  1895,  p.  299-300. 


LUNAR  AND  TERRESTRIAL  ALBEDOES     31 

If  the  sum  total  of  reflected  rays  of  every  wave-length  agrees 
approximately  with  unaltered  solar  radiation,  the  preceding  frac- 
tion must  be  increased  a  little  to  represent  the  result  as  it  would 
be  found  outside  the  earth's  atmosphere ;  because  the  solar  reflected 
rays  are  of  shorter  wave-length  than  the  rays  emitted  by  the  moon, 
and  they  are  differently  modified  in  passing  through  the  at- 
mosphere, which  alters  the  relative  values  of  the  terms  of  the 
comparison.  Except  for  certain  bands  of  selective  absorption,  the 
longer  waves  are  more  readily  transmitted  by  the  air.  It  becomes 
increasingly  evident  that  the  solar  constant  is  about  3.5  (C.  G. 
Min.),  but  this  is  reduced  to  1.5  at  sea-level,  so  that  the  real  trans- 
mission of  solar  rays  by  the  atmosphere  is  3/7.  In  a  seasonally 
comparable  observation  of  the  moon,  48  per  cent,  of  its  emitted 
radiation  entered  through  the  air.  Reducing  to  conditions  outside 
the  atmosphere  by  these  values, 

radiation  reflected  by    the    moon          !6.8X(7/3)  I 

radiation  absorbed  by  the  moon  83.2  X  (I(V4.8)  ~  4.42 
and  the  true  percentage  of  total  solar  radiation  reflected  from 
the  moon  is  100/5.42  =  18.5%,  which  differs  little  from  a  mean  of 
the  three  results  quoted  for  the  reflected  light  of  the  moon, 
A.,=  18.196-  These,  however,  as  I  shall  show,  need  to  be  dimin- 
ished somewhat. 

The  question  whether  the  invisible  and  longer  solar  waves  of 
radiation  are  better  or  worse  reflected  by  the  moon  than  the  visible 
ones  has  never  been  definitely  settled,  and  indeed  there  is  diversity 
of  opinion  as  to  the  relative  reflection  by  the  moon  of  different 
colors  in  the  visible  spectrum.  We  need  not  consider  the  great 
bands  of  "metallic"  reflection  by  quartz  near  9/x  and  the  large 
reflection  by  many  common  terrestrial  substances  between  8  and 
lOfji,  for  there  is  very  little  solar  radiation  of  these  wave-lengths 
to  suffer  reflection.1  Metals  have  greater  specular  reflection  for 
infra-red  radiation  just  beyond  the  visible  spectrum  than  for  lu- 
minous rays ;  but  metals  are  not  in  question  here.  The  lunar  reflec- 
tion is  almost  entirely  diffusive,  and  we  wish  to  know  how  sub- 
stances which  reflect  diffusely  behave  to  infra-red  rays  between 
0.7  and  3-O/u..  Eighteen  years  ago,  I  published  the  value  of  13.1 

1  See  my  paper  on  "The  Temperature  Assigned  by  Langley  to  the 
Moon,"  Science,  N.  S.,  Vol.  XXXVII,  No.  964,  pp.  949-957,  June  20, 
1913. 


LUNAR  AND  TERRESTRIAL  ALBEDOES 


per  cent,  for  the  lunar  reflection  of  total  solar  radiation,1  but  I 
now  think  that  this  should  be  increased  to  the  value  given  above, 
(-18.5%),  because  in  my  former  work  I  under-rated  the  absorp- 
tion of  solar  radiation  by  the  air. 

On  the  other  hand,  on  the  strength  of  Zollner's  oft  quoted,  but 
little  studied  value  of  what  he  calls  the  "true"  lunar  albedo 
(17.4%)  I  had  formerly  supposed  that  luminous  rays  are  better 
reflected  than  the  visible  ones  from  0.7  to  3-O/x;  but  it  now  appears 
probable  from  measures  which  are  to  follow,  that  this  relation 
must  be  reversed,  and  that  the  larger  luminous  reflection  which 
would  result  from  the  lunar-solar  ratios  adopted  by  Miiller  and 
Russell,  can  not  be  accepted.  In  fact,  in  place  of  Zollner's  hitherto 
accepted  albedo  must  be  substituted  the  smaller  value  A2  =  o.i$(), 
which  follows  from  his  own  observations  (entirely  apart  from  any 
considerations  whatsoever  as  to  the  shape  of  the  moon,  or  as  to  its 
surface  quality,  or  the  peculiarities  of  its  phase  law). 

I  will  now  give  a  series  of  ratios  of  sunlight  to  moonlight  for 
homogeneous  radiations  in  the  visible  spectrum  derived  from  my 
spectro-photometric  observations,  published  in  Astronomische 
Nachrichtcn,  Nr.  4820  (s.  385 — 386)  which,  as  there  given,  are 
corrected  for  atmospheric  absorption  only.  The  original  values 
are  all  that  is  necessary  for  a  comparison  of  the  relative  reflection 
of  different  colors  by  the  moon,  but  for  our  present  purpose  they 


I 

Sunlight 

A 

Sunlight       2 

Moonlight 

Moonlight 

p- 

/* 

0.40 

531,000 

0.56 

682,000 

0.42 

559,000 

0.58 

666,000 

0.44 

587,000 

0.60 

646,000 

0.46 

607,000 

0.62 

619,000 

0.48 

640,000 

0.64 

600,000 

0.50 

669,000 

0.66 

513,000 

0.52 

681,000 

0.68 

456,000 

0.54 

685,000 

Mean  = 

609,000 

1  Astrophysical  Journal,  Vol.  VIII,  p.  275,  December,  1898. 

2  The  reduction  factor  to  moon's  mean  distance  from  the  earth, 
and  to  full  moon,  rests  on  the  following  data : 

Series  1                Series  2  Series  3 

Mean  phase-angle  from  full,  a  —  —  30°           a  =  —  18°  a  =  —  6° 

Light  (from  lunar  phase-curve),        0.53                       0.69  0.90 

Moon's  parallax,                                   3374"                    3415"  3455" 

Reduction  factors,                                0.515                      0.687  0.916 


LUNAR  AND  TERRESTRIAL  ALBEDOES     33 

require. further  reduction  for  the  distance  of  the  moon  and  for  the 
interval  to  exact  full  moon.  The  original  figures  have  been 
multiplied  by  the  factor  0.70  and  are  fully  corrected.  The 
spectro-photometric  method  yields  results  which  have  one  special 
advantage.  They  are  entirely  free  from  the  troublesome  Purkinje 
effect  which  has  vitiated  much  of  the  previous  measurement. 

The  mean  ratio  of  sunlight  to  moonlight  for  light  of  every  color 
within  the  visible  spectrum  is 

sunlight    :  moonlight  =  609,000    :  i, 

but  considering  that  the  central  region  in  the  green  affects  the  eye 
most  powerfully,  a  mean  visual  ratio  of  681,000  :  I  is  to  be 
preferred. 

It  is  evident  from  inspection  of  the  numbers  in  this  table  that, 
while  the  reflection  of  blue  and  violet  light  by  the  moon  is  larger 
than  that  in  the  green  and  yellow,  there  is  also  a  large  reflection 
in  the  red  which  increases  in  the  direction  of  the  infra-red. 
Unfortunately,  there  is  a  dearth  of  observations  of  the  reflective 
power  of  ordinary  terrestrial  materials  in  the  region  between  0.7 
and  3-O/x,  where  there  is  a  great  block  of  solar  radiant  energy ; 
but  I  think  we  may  conclude  that  this  region  is  probably  more 
reflected  by  the  moon  than  the  visible  part  of  the  spectrum.  In 
this  case,  Russell's  ratio  (i  :  465,000)  may  answer  well  enough 
for  the  reflection  of  solar  infra-red  radiation  by  the  moon,  but  it 
is  much  too  large  for  the  reflection  of  visible  rays. 

The  earth  can  not  have  the  same  ratio  of  reflection  for  visible 
and  infra-red  rays  that  the  moon  does,  because  the  earth's  reflec- 
tion is  mainly  atmospheric,  with  visible  rays  somewhat  better 
reflected  than  the  infra-red.  If  the  moon's  geometrical  albedo 
for  visible  rays  is  Amz  =  0.15,  that  of  the  earth  is 
Ae2  --  4.8  X  ai5  =  O-72-'  but  the  reflection  of  total  radiation 
by  the  earth,  unlike  that  by  the  moon,  is  smaller  than  for  visible 
rays  (because  the  infra-red  rays  are  but  little  reflected  by  the 
atmosphere).  It  can  hardly  exceed  Ae2w  =  0.70,  and 
may  be  as  low  as  0.50.  For  the  present  I  shall  adopt  Ae2(l)  =0.60. 
Whatever  values  are  finally  adopted  ought  to  be  consistent  among 
themselves  and  with  the  general  principles  now  under  discussion. 

In  my  visual  photometric  work  on  the  earth-shine,  the  intrinsic 
brightness  of  the  moon  at  quadrature  is  taken  =  0.16  times  the 
light  at  full  moon,  giving 


34     LUNAR  AND  TERRESTRIAL  ALBEDOES 

full-moon  light:  full-earth  light  =  1,600  :  0.16  =  10,000  :  i, 
since  I  found  that  full  earth-shine  at  the  time  of  new  moon 
must  be  about  1/1,600  of  the  light  of  a  corresponding  sun- 
lit area  of  the  moon  near  quadrature.1  It  was  not  neces- 
sary for  this  purpose  that  the  whole  illuminated  surface  of 
the  moon  should  be  measured,  nor  is  it  possible  to  make  such  a 
measure  of  the  earth-shine  directly  at  new  moon  ;  but  it  is  essential 
that  the  lunar  surfaces  to  be  compared  shall  be  similar,  and 
Professor  Russell's  arbitrary  change  of  my  mean  ratio  is  not  ad- 
missible, even  after  granting  the  large  probable  error  of  the 
result.2 

I  propose  to  give  equal  weights  to  the  results  of  my  own 
measures  and  to  those  of  Zollner  as  now  correctly  reduced.  Com- 
paring the  lunar-solar  ratio  with  the  moonlight :  earth-shine  ratio 
given  above,  we  have  (with  Very's  value) 

sunlight   :  full-earth  light  =  681,000   :  10,000  —  68.1    :  i, 
(with  Zollner's  value) 

sunlight   :  full-earth  light  =  618,000  :  10,000  =  61.8   :  i. 
Allowing  for  the  greater  area  of  the  earth  as  seen  from  the  moon, 
these  ratios  become : 

68.1/13.4  =  5.08,  and  61.8/13.4  =  4.61,  mean  =  4.8. 
Moon's  geometrical  albedo   (for  the  visual  effect) 

According  to  Very,  Am2  =  98,317/681,000  =  0.144  "] 

A          ,-  •         ^..,,  [mean=0. 15 

According  to  Zollner,  /4m2=98,3i7/6i8,ooo=:o.i59  J 

Earth's  geometrical  albedo  =  4.8  X  o>15  —  0.72 
Geometrical  reflection  of  total  radiation : 

Moon  =  0.185,    Earth  =  0.60   (?) 

I  take  the  reduction  factor  for  spherical  albedo,  q  =  0.35  for  the 
moon  from  the  integration  of  its  phase-curve,  and  twice  this,  or 
9  =  0.70  for  the  earth,  which  is  a  little  less  than  the  Lambert 
value. 

Spherical  albedo  of  moon,  Ami  =  0.35  X  0.15  =  0.053 
Spherical  albedo  of  earth,  /4ei  =  0.70  X  °-72  =  0.504. 

1  Astronomische  Nachrichten,  Nr.  4696,  s.  286. 

2  With  the  increased  assurance  given  by  the  good  agreement  of 
the  photographic  result,  I  do  not  believe  that  this  error  can  amount  to 
as  much  as  10%.     Some  weighting  of  the  observations   is  perhaps 
desirable. 


LUNAR  AND  TERRESTRIAL  ALBEDOES     35 

Moon's  Stellar  Magnitude: 

Difference  of  magnitude  from  sun  =  log  681,000/0.4  =  -j-  14.58 
Stellar  magnitude  of  sun  (Russell)  —  —  26.72 

Stellar  magnitude  of  moon  (Very)  =  —  12.14 

Photographic  magnitude  of  moon    (King)  -  11.37 

(with  Russell's  phase-curve,  etc.) 

Moon's  Color-Index    (King- Very)  =-{-0.77, 

or  a  little  less  than  that  of  the  sun,1  which  agrees  with  Abney's 
photographic  observations,  confirmed  by  my  spectro-photometric 
comparison  of  sun  and  moon,  in  showing  that  the  moonlight  is 
bluer  than  sunlight.  I  have  shown  that  the  moonlight  is  redder 
than  sunlight  in  the  extreme  red,  but  these  rays  do  not  count  for 
much  either  photographically  or  visually,  and  therefore  do  not 
effect  the  color-index  as  usually  defined. 

Earth's  Stellar  Magnitude  (Very)  : 

As  seen  from  moon,  -12.14  —  4- 58  =  — 16.72 

As  seen  from  sun,  —16.72-}-  12.95=  —  3-77 

I  make  no  further  claim  for  my  previously  published  value  of 
the  earth's  albedo2  than  that  its  ratio  to  the  moon's  albedo  has 
been  fairly  well  determined.  The  previous  figures  were  based 
upon  Zollner's  published  lunar  albedo  and  must  be  diminished  a 
little  according  to  what  precedes ;  but  this  will  not  effect  the  argu- 
ment for  a  high  value  of  the  solar  constant,  because  this  rests  on 
wholly  different  grounds.  If  I  could  accept  in  principle  Professor 
Russell's  argument  that  my  measurement  of  the  earth-shine,  "far 
from  being  inconsistent  with  Abbot's  value  of  the  solar  constant 
(1.93  calories)  is  actually  in  agreement  with  it,"3  since  it  has 
now  been  shown  that  Russell's  p  must  be  doubled  to  give  the 
geometrical  albedo,  I  might  claim  that  Abbot's  constant  should 
be  doubled !  But  unhappily  this  simple  method  of  disposing  of  the 
solar-constant  problem  will  not  work.  A  high  value  of  terrestrial 
reflection  of  total  solar  radiation  is  indeed  inconsistent  with  a  low 
value  of  the  solar-constant,  but  a  low  value  of  this  reflection  is  not 
necessarily  inconsistent  with  a  high  value  of  solar  radiation, because 
the  atmospheric  depletion  of  the  sun's  rays  is  composed  of  several 

1  Color-index  of  sun  =  between  +  0.8  and  +  0.9,  that  of  Capella 
being  +  1.0. 

2  Astronomische  Nachrichten,  Nr.  4820,  s.  400. 

3  Astrophysical  Journal,  April,  1916,  p.   195. 


36  LUNAR  AND  TERRESTRIAL  ALB E DOES 

parts.  If  reflection  is  found  to  be  less  potent  than  has  been  sup- 
posed, this  simply  puts  a  heavier  burden  on  the  other  processes  of 
depletion. 

I  have  elsewhere  concluded1  that  only  about  18  per  cent,  of 
the  sun's  rays,  received  upon  the  entire  sunward  hemisphere  of  the 
earth,  are  effective  at  the  earth's  surface  in  production  of  tem- 
perature. Out  of  the  82  per  cent,  of  solar  radiation  lost  by  the 
sunward  hemisphere  of  the  earth  in  one  way  or  another  in  pas- 
sing through  the  earth's  atmosphere,  the  measurement  of  the 
earth's  spherical  albedo  which  has  just  been  given  indicates  that 
approximately  50  are  reflected  back  to  space  by  air,  or  by  clouds, 
including  a  reflection  of  a  few  per  cent,  by  the  solid  or  liquid  sur- 
face of  the  earth.  The  rest  of  the  depletion  is  divided  among 
agencies  which  go  under  the  general  name  of  "absorption,"  but 
this  also  is  really  a  complex  of  several  processes. 

As  was  pointed  out  in  my  "Note  on  Atmospheric  Radiation,"2 
a  portion  of  the  incoming  solar  radiation  of  short  wave-length  is 
used  up  in  the  upper  air  in  ionization  of  atmospheric  ingredients,  or 
in  the  production  of  ozone  and  other  highly  efficient  absorbents; 
and  since  there  is  at  present  no  way  of  finding  out  how  potent 
this  part  of  the  atmospheric  process  may  be  (except  possibly 
through  an  interpretation  of  certain  little  understood  facts  made 
known  to  us  in  the  study  of  atmospheric  thermodynamics)  it  is 
possible,  as  Dr.  Louis  Bell  has  suggested  to  me,  that  more  solar 
energy  than  we  imagine  is  lost  in  the  ionization  processes ;  and  in 
this  case  quite  a  little  of  the  remaining  32  per  cent,  of  "absorp- 
tion," so-called,  may  be  ionization  by  solar  radiation  of  very  short 
wave-length  at  great  altitudes  in  the  atmosphere,  these  rays  being 
wholly  obliterated  in  the  process. 

I  have  alluded  to  the  changes  which  Professor  Russell  has 
introduced  into  my  earth-shine  measures  as  founded  on  mis- 
apprehensions, and  must  now  substantiate  this  claim.  On  page 
185  of  his  second  article  we  are  told  that  "Table  IVA  contains  data 
derived  from  Very's  paper" ;  but  the  mode  of  derivation  is  not 
consistent.  The  first  three  numbers  in  the  last  column  have  been 
obtained  by  multiplying  the  ratio  of  exposure-durations  for  earth- 
shine  and  for  sunlit  moon  by  the  ratio  of  photographic  intensities ; 

1  Astrophysical  Journal,  Vol.  XXXIV,  p.  382,  Dec.,  1911. 

2  American  Journal  of  Science,  Vol.  XXXIV,  p.  533,  Dec.,  1912. 


LUNAR  AND  TERRESTRIAL  ALBEDOES     37 

but  the  last  two  numbers  have  been  found  by  dividing  the  first 
quantities  by  the  second.  By  this  means,  the  two  sets  of  numbers 
(for  January  and  August  respectively)  which,  if  they  had  been 
correctly  derived,  would  have  been  entirely  different,  because  as 
yet  uncorrected  for  photographic  peculiarities,  are  brought  into 
seeming  approximate  agreement,  and  the  conclusion  is  reached 
that  the  photographic  correction  which  I  have  derived  for  the  rel- 
atively over-exposed  lunar  spectrograms  was  unnecessary,  and  that 
my  corrected  values  are  wrong !  By  this  wholly  erroneous  argu- 
ment, Russell  supports  his  reduction  of  my  ratio  of  the  earth's 
albedo  to  the  moon's  from  the  spectrograms,  to  a  quantity  about 
half  as  great  as  mine.  It  is  needless  to  say  that  the  seeming  agree- 
ment of  the  numbers  in  the  last  column  of  Table  IVA  is  wholly 
accidental. 

On  page  184  (op.  cit.}  Professor  Russell  says  that  my  "con- 
clusions regarding  the  relative  intensity  of  the  light  of  these  two 
sources  [the  earth-lit  and  the  sun-lit  portions  of  the  moon]  depend 
on  assumptions  regarding  the  photographic  action  of  exposures  to 
light  of  different  brightness."  On  the  contrary,  my  results  do  not 
rest  on  "assumptions,"  but  on  carefully  executed  quantitative 
measurements  of  the  photographic  effects  in  question  throughout 
the  entire  visible  spectrum.  When  properly  reduced,  there  is  no 
difference  between  the  results  of  the  photographic  and  of  the 
visual  observations.  The  statement  (op.  cit.,  p.  186)  that  "the 
photographic  observations  therefore  make  the  earth-shine  only 
half  as  bright  as  do  the  visual  observations,"  is  consequently  en- 
tirely wrong;  and  the  conclusion  that  "this  is  just  what  might  be 
expected  if  the  plates  had  followed  the  ordinary  law  for  faint 
illumination  and  long  exposure,  and  been  'less  sensitive  than 
i  X  t' >"  is  equally  erroneous.  The  error  has  come  from  the  in- 
correct reduction  of  my  observations  by  Russell  in  the  aforesaid 
Table  IVA. 

The  discrepancy  which  Professor  Russell  thinks  he  finds 
between  my  theory  and  my  observations  in  connection  with  the 
phase-curve  of  the  moon  (op.  cit.,  p.  186  to  187)  does  not  really 
exist.  The  observations  had  first  to  be  reduced  to  a  constant  unit 
of  comparison  surface,  and  then  to  a  selected  lunar  phase-angle. 
As  it  happened,  the  two  corrections  in  a  particular  case  were  of 
equal  numerical  value,  but  opposite  sign.  Limited  areas  of  the 


38     LUNAR  AND  TERRESTRIAL  ALBEDOES 

moon  were  necessarily  observed,  and  the  comparison  in  every  case 
in  the  visual  measures  was  between  "twin  circular  apertures  in 
black  card,  each  subtending  7'  of  arc  on  the  celestial  sphere,  one 
above,  and  the  other  below  the  horizontal  line  of  junction  of  the 
last  pair  of  reflecting  prisms."  1  One  of  these  apertures  was  illu- 
minated by  the  standard  light,  and  the  other  by  a  sample  of  the 
lunar  surface.  In  preparing  the  material  of  A. N.  4696  for  the  press, 
I  have  omitted  a  remark  which  is  needed  to  complete  the  sense. 
After  the  first  equation  at  the  top  of  page  286  in  that  paper,  there 
should  be  inserted  these  words :  "The  ratio  of  intrinsic  brightness 
for  the  particular  (limited)  region  of  the  moon  under  observation 
is  the  inverse  of  that  just  given."  A  quotation  from  my  note  book 
will  clear  up  the  matter  fully:  "At  cp  =  44°,  M  =  0.090. 
At  (p  =  87°,  M  =  0.058.  Ratio  —  1.55  :  i.oo.  For  equal  moon- 
light E/M  (for  (p  •=  44°)  must  be  multiplied  by  1.55."  The  phase- 
reduction  required  division  by  the  same  number.  This  peculiarity 
is  due  to  an  exceptionally  large  reflection  from  the  lunar  substance 
when  the  angle  of  incidence  is  large  and  nearly  equal  to  the  angle 
of  reflection,  as  was  the  case  for  the  particular  lunar  region 
observed  with  the  moon's  elongation,  qp=44° ;  but  the  reflection 
diminished  as  the  angle  of  incidence  of  the  solar  rays  on  this  spot 
decreased.  The  Lommel-Seeliger  law  was  formulated  to  deal  with 
just  such  peculiarities.  My  published  result: 

"E44   :  E87=  (0.0003477  X  i -55)    :  0.0002210 

=  0.0005389   :  0.0002210 
-2.438  :  i," 

(op.  cit.,  p.  286)  still  stands  and  is  fairly  comparable  with  the 
ratio  for  Venus,  computed  by  me  from  Miiller's  result,  namely, 

V44   :  V87  =  2.100   :  i, 
or  as  Russell  gives  it  for  elongations  30°  and  90°, 

V30  :  V90  =  2.74  :  i.oo, 

where  Lambert's  law  would  give  2.94   :  i.oo.     The  agreement  is 

close  enough  to  show  that  the  phase-law  for  the  earth  resembles 

that  for  Venus,  approaching,  however,  a  little  more  nearly  to 

the  Lambert  law,  and  is  quite  different  from  that  for  the  moon. 

On  page  189,  Professor  Russell  says :  "Very's  observations 

of  the  earth-shine  indicate  that  the  mean  full  earth,  as  seen  from 

1  Astronomische  Nachrichten,  Nr.  4696,  s.  269. 


LUNAR  AND  TERRESTRIAL  ALBEDOES     39 

the  moon  is  forty  times  brighter  than  the  full  moon  as  seen  from 
the  earth,"  where  the  forty  should  be  sixty-eight. 

Russell's  adopted  value  of  the  moon's  stellar  magnitude 
( — 12.55)  's  °-4I  magnitude  brighter  than  mine  ( — 12.14). 
Zollner's  result  from  a  comparison  with  the  sun  gave  the 
intermediate  value, —  12.24,  while  that  from  his  Capella  com- 
parison, — 12. 18,  approaches  still  more  nearly  to  mine.1 

The  last  line  of  Russell's  Table  V.  (op.  cit.,  p.  190)  which 
purports  to  give  "the  earth  (from  Very's  reductions  of  Slipher's 
spectrograms)"  is  misleading,  since  he  has  substituted  his  own 
reduction  for  mine. 

A  final  word  may  be  permitted  on  the  vicissitudes  of  the  solar- 
lunar  light-ratio.  Bouguer,  who  obtained  a  ratio  of  300,000  :  I 
for  sunlight  to  moonlight  (Traite  d'Optique,  p.  87,  1760)  was 
careful  to  observe  when  the  full  moon  was  near  its  mean  distance 
and  when  both  bodies  were  at  the  same  altitude ;  but  unfortunately, 
he  thought  it  necessary  to  use  identical  optical  means  in  either 
case,  and  therefore  his  candle  had  to  be  at  a  distance  of  50  feet 
for  the  moon  and  i^  feet  for  the  sun,  so  that  the  illuminations 
actually  measured  were  in  the  ratio  of  1407  :  I,  both  moonlight 
and  sunlight  having  been  much  reduced.  Under  these  cir- 
cumstances, the  bluer  light  of  the  heavenly  bodies  being  compared 
with  reddish  candle  light,  the  moonlight,  on  account  of  its  greater 
faintness  and  of  the  relatively  greater  sensitiveness  of  rod-vision 
for  faint  blue  light  as  the  general  illumination  diminished,  had  an 
undue  advantage,  in  the  candle  comparison,  over  sunlight,  as  will 
be  evident  from  a  short  table  in  my  paper  on  "The  Earth's 
Albedo."  2  Bouguer's  300,000  must  be  at  least  doubled  to  correct 
for  this  error.  The  spectrophotometric  method  entirely  removes 
this  difficulty. 

1  From  Zollner  (op.  cit.,  p.  125  and  p.  105). 

Log  ratio  Sun  :  Capella          =     10.7463 
Log  ratio  Sun  :  Moon  =       5.7910 


Log  ratio  Moon  :  Capella        =       4.9553 
Log  ratio  divided  by  0.4  -  12.39 

Stellar  magnitude  of  Capella  =  +     0.21 


Stellar  magnitude  of  Moon     =  —  12.18 
2  Astronomische  Nachrichten,  Nr.  4696,  s.  276. 


40     LUNAR  AND  TERRESTRIAL  ALBEDOES 

Dr.  W.  H.  Wollaston,  who  found1  that  the  sun  = 

5,563  X  ( 12)  2  =  801,072  moons, 

used  better  observing  conditions,  but  he  made  only  two 
readings  on  the  moon.  In  one  observation  at  full,  his 
candle  was  placed  at  12  feet.  In  the  other,  made  at  a  time 
when,  if  the  atmosphere  had  been  equally  transparent  the  light 
should  have  been  0.84  of  that  at  full  moon,  the  same  candle- 
reading  "12  feet,"  was  recorded.  One  can  not  help  surmising  that 
neither  reading  was  better  than  a  rough  approximation. 

Bond's  solar-lunar  ratio,  471,000  :  I,  is  an  underestimate  for 
the  same  reason  that  Bouguer's  value  is  too  small.  Zollner's 
criticism  of  Bond's  fireworks  as  not  accurate  enough  for  standards 
is  also  fully  justified. 

Zollner's  own  measurements  of  the  ratio  of  sunlight  to  moon- 
light appear  to  have  been  made  with  great  care ;  but  in  reducing 
them  he  becomes  lost  in  the  mazes  of  an  unnecessarily  complex 
argument.  With  the  removal  of  this  blemish,  no  fault  can  be 
found  with  the  new  value  deduced  from  the  original  measures. 
The  same  can  not  be  said  of  Zollner's  isolated  measurements  of 
the  earth-shine-  which  require  unknown  corrections  for  skylight. 
The  earth-shine  observations  of  Arago  and  Laugier  have  been 
utilized  by  me  in  conjunction  with  my  own  with  which  they  are 
in  good  agreement. 

Various  other  more  or  less  aberrant  values  of  the  moon's 
albedo  usually  err  from  inadequate  correction  for  changes  in 
atmospheric  transparency.  As  an  instance  of  a  great  name  attached 
to  an  extraordinarily  small  value  which  is  simply  impossible,  may 
be  cited  that  of  William  Thomson  (Lord  Kelvin)  :  "70,000  :  i" 
for  the  ratio  of  sunlight  to  full-moon  light.2 

Whatever  faults  may  still  remain  in  the  values  which  are 
given  here,  they  at  least  have  this  merit,  that  they  are  consistent 
among  themselves,  which  is  very  far  from  being  the  case  with 
the  results  which  have  been  published  hitherto. 
WESTWOOD  ASTROPHYSICAL  OBSERVATORY, 
August,  /o/(5. 

1  Philosophical  Transactions  of  the  Royal  Society  of  London,  Vol. 
CXIX,  p.  19-27,  1829. 

2  Poggendorff's  Jubelband,  p.  624,  1874. 

3  Nature,  Vol.  XXVII,  p.  279,  January  18,  1883. 

THE  LIBRARY 


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